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LIBRARY OF THE University of California. Class |
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CONTENTS.
I. Algebra.
Preliminary Notions
Definitions and Symbols
Addition
Subtraction .
Brackets — Vincula
Introductory Simple Equations
Questions in Simple Equations
Definitions continued
Multiplication
Division
Algebraic Fractions
Addition — Subtraction
Multiplication — Division
Grejitest Common Measure
Least Common Multiple .
Simple Equations in general
Simultaneous Simple Equations
Theory of Exponents
Square Root of Poljmomial
Cube Root of Polynomial
Surds — Imaginary Quantities
Binomial Surds
Operations with Imaginary Quanti
ties .... Quadratic Equations, Rule I.
,, ,, Rule II.
Theory of Quadratic Equations Simultaneous Quadratics Theory of Proportion Variation ... Arithmetical Progression Geometrical Progression . Harmonical Progression . Piling of Balls and Shells Square Pile — Rectangular Pile General Rule for all Piles The Binomial Theorem . Limitations of the Theorem The Exponential Theorem Theory of Logarithms , Construction of Logarithms Exponental Equations Compound Interest Annuities
Increase of Population Permutations Combinations Probabilities . Life Annuities Life Assurances
PAGE
1
2 4 6 7 8
10 12 13 16 19 20 21 22 24 25 30 36 37 39 40 42
45, 46 49 50 52 58 60 61 62 64 67 68 69 70 77 78 79 80 84 85 86 87 88 91 93 99 102
Theory of Equations Transformation of Equations Limits of the Roots Equal Roots . Rule of Descartes . Criteria of Imaginary Roots Newton's Rule
Theorem for the Biquadratic , Solution of Equations . Newton's Approximation Approximation by Position Cardan's Method for Cubics Decomposition of a Biquadrati Recurring Equations Binomial Equations Vanishing Fractions Maxima and Minima Indeterminate Equations Indeterminate Coefficients Summation of Finite Series The Diflferential Method Construction of Tables . Interpolation
Summation of Infinite Series Recurring Series . Reversion of Series Convergency of Series
II. Plane TRiaoNOMETRV.
Definitions, &c. Measurement of Angles . Sine, Cosine, Tangent, &c. Fundamental Relations . Solution of Right-angled Triangles Oblique-angled Triangles Miscellaneous Problems . Quadrature of the Circle Unit of Circular Measure French and English Degrees . Extension of Definitions Sine and Cosine of (A+B) Expressions involving Two Angles Single Angles and their Halves Multiple Angles — Ambiguities Applications of Formulai Inverse Trigonometric Functions Solution of a Quadratic by Tables Solution of a Cubic by Tables Construction of Trigonometric Tables Developments of sin ^, cos ^ .
CONTENTS.
Euler's Expressions for sin 6^ cos 6 . De Moivre's Theorem Developments of cos"^, sin"^ . Developments of sinw^, cos?i^ . Development of ^ in powers of tan ^ Euler's Series : Machin's Series Developments of y, of sin fi, and of
cos ^
Wallis's Expression for ^w Imaginary Logarithms The Numbers of Bernoulli Summation of Trigonometric Series Imaginary Roots of Unity Construction of Log Sines and Co-
page 213 213 215 216 216 217
218 218 219 220 222 225
226
III, Spherical Trigonometry.
Preliminary Theorems . . . 227 Fundamental Formulae . . . 231
Formulae for Sides . . . .233 Napier's Analogies . . . 235
Right-angled Triangles : Napier's
Rules 236
Quadrantal Triangles . . . 238 Solution of Oblique-angled Triangles 240 Examples of the Six Cases . 240-250 Area of Triangle : Spherical Excess . 252
IV. Mensuration.
Area of Parallelogram and Triangle ,, ,, Triangle : three sides given Areas of Quadrilaterals . Equidistant Ordinates . Regular Polygons .... The Circle and its Sectors Segment of a Circle Circular Ring ....
Inscribed and Circumscr. Triangle . Prism and Cylinder Pyramid and Cone .... Frustum of Pyramid or Cone . Surface of a Frustum The Sphere and its Surface Theorem of Archimedes . Volume of Segment and Zone . Equidistant Sections . . .
Weight, &c., of Shot and Shells . Weight of Powder in Shells . ^. Applications of Maxima an^l Minima
V. Analytical Geometry.
Equations of a Point Equation of a Straight Line . Straight Line subject to Conditions Problems on the Straight Line The Circle : Rectangular Axes The Circle : Oblique Axes Tangent at a Point Tangent from a Point
255
257
258
260
261
262
264
265
266
267
268
269
270
271
279-
273
274
275
277
279
285 287 291 296 300 301 302 302
Locus of an Equa. of Second Degree
Construction of Loci
Problems on the Circle .
Polar Co-ordinates
The Conic Sections
Equation of the Ellipse .
The Principal Diameters
Change of Co-ordinates .
Conjugate Diameters
Change from Oblique to Rectangular Co-ordinates ....
Tangent to the Ellipse .
Subtangent, Normal, Subnormal
Polar Equation of the Ellipse .
Radius of Curvature
Chord of Curvature
Area of the Ellipse
The Hyperbola ....
Equation of the Hyperbola
Principal Diameters
Conjugate Diameters
Tangent to the Hyperbola
The Asymptotes ....
Equation with Asymptotes for Axes
Conjugate Hyperbolas
The Parabola ....
Parabola referred to Conjugate Axes
Tangent, Normal, &c. .
Properties connected with Tangents .
Polar Equation of the Parabola
Area of the Parabola . " .
Radius of Curvature
Locus of Equation of the Second De- gree ......
The Different Curves represented
Determination of particular Loci
Problems on Loci ....
VI. Mechanics : Statics.
Conspiring and Opposing Forces
The Parallelogram of Forces
The Triangle of Forces .
Composition of Forces .
The Polygon of Forces .
Problems in Statics
The Principle of Moments
Parallel Forces acting in a Plane
Centre of Gravity of a Rigid Body
General Equations of Parallel Forces
Equilibrium in general .
Problems on Equilibrium
Mechanical Powers
Lever — Balance — Steelyard
Combination of Levers .
Wheel and Axle : Toothed Wheels
Pulley : Systems of Pulleys .
Inclined Plane : Screw .
The Wedge .
Mechanical Powers in Motion .
Principle of Virtual Velocities
Friction ....
304 305 306 309 310 311 313 315 316
318 319 322 326 327 328 329 330 331 333 334 336 336 339 340 341 342 344 345 348 349 350
351 354 355 357
361 362 364 365 367 367 372 375 377 383 387 390 396 396 401 402 404 407 410 412 418 410
CONTENTS.
XI
VII. Dynamics.
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Uniform Motion .... |
424 |
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Variable Motion .... |
425 |
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Accelerated Motion |
426 |
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Falling Bodies : Gravity- |
429 |
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Motion on Inclined Planes |
430 |
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Parallelogram of Velocities |
431 |
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Motion of Projectiles |
432 |
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Circular Motion : Centrifugal Force . |
439 |
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Moving Forces .... |
442 |
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Principle of D'Alembert |
449 |
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Moment of Inertia |
450 |
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Impact of Bodies . . . . |
451 |
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VIII. Hydrostatics. |
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Transmission of Pressure |
456 |
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Hydrostatic Paradox |
457 |
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Levelling Instrument |
460 |
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Explanation of Symbols, &c. . |
460 |
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Centre of Pressure |
465 |
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Kesultant of Fluid Pressures . |
466 |
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Equilibrium of a Floating Body |
467 |
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Equilibrium of a Rotating Fluid |
469 |
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Specific Gravities of Bodies . |
470 |
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Hydrostatic Balance . . |
472 |
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Common Hydrometer |
472 |
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Nicholson's Hydrometer . |
473 |
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Elastic Fluids : the Atmosphere |
476 |
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Law of Mariotte and Boyle . |
477 |
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Altitudes by the Barometer . |
479 |
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The Wheel Barometer . |
482 |
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The Thermometer . . . . |
482 |
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The Syphon |
482 |
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The Common and Forcing Pumps . |
483 |
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The Diving Bell . . . . |
485 |
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Air Pumps . . . . . |
486 |
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The Condenser . . . . |
487 |
IX.
Differential and Integral Calculus.
Differentiation .... 489 Investigation of Rules . . . 494 Algebraic Functions . . . 495 Applications to Geometry . . 497 Log. and Exp, Functions . . 498 Trigonometrical Functions . . 600 Inverse Functions . . . .600 Integration of Particular Forms . 603 Areas of Curves : Definite Integrals 610 Volumes of Revolution . . .613
Lengths of Curve Lines . Surfaces of Revolution , Successive Differentiation Theorem of Leibnitz Maclaurin's Theorem Taylor's Theorem .... Limits of Taylor's Theorem
,, ,, Maclaurin's Theorem Compound Functions Implicit Functions Vanishing Fractions Maxima and Minima Max. and Min. of Implicit Functions Functions of Two Variables . Change of Independent Variable Failure of Taylor's Theorem .
,, ,, Maclaurin's Theorem Asymptotes to Curves Spiral Curv^es Circular Asymptotes Sectorial Areas Contact : Osculation Rad. of Curv. in Rect. Co-ordinates Rad. of Curvature in Polar Curves . Chord of Curvature Consecutive Lines and Curves Enveloping Curves Singular Points .... Integ. of Rational Fractions .
,, ,, Irrational Functions
Integration by Reduction
,, ,, Parts
,, ,, Series
Successive Integration .
£14 516 619 620 621 625 627 529 630 633 634 641 645 548 652 653 655 655 658 660 661 663 665 568 669 673 576 576 581/ 685 687 690 694 696
X. Applications to Mechanics.
The Centre of Gravity . . .597
Theorems of Guldinus . . .699
Moment of Inertia : Gyration . 600
Centre of Oscillation . . .603
Centre of Percussion . . . 604
Attraction of Bodies . . . 605
Velocity : Acceleration . . .608
Equations of Motion . . .609
Cycloidal Pendulum . . . 613 Simple Pendulum .... 614
Problems on the Pendulum . . 615
The Ballistic Pendulum . . .618
The Cycloid 619
The Catenary . . . .620
Note on Interpolation . . . 622
Answers to Examples . . . 624
ERKATA.
Page 163, Ex. 8, for = read +. P. 183, line 2, for 88 read 80 ; and line 4, for 33 read 25. P. 206, Ex. 17, for tan read sin ; and Ex. 19, for tan b read tan B. P. 381 in the diagram, the point e should be on OE. P. 446, line 18, for exactly read very nearly. P. 449, line 22, after denominator put -^g. P. 571, last line but one, omit comma after C.
Types fallen out or defective. Page 27, Ex. 10, -. P. 64, line 6, the exponent n-fjp.
7) 25
P. 72, Une 8, -. P. 77, line 19, — . P. 169, line 29, the minus sign. q 2
P. 224, line 1, the exponent w. P. 225, line 16, the exponent .
n
P. 297, Prob. II., |. P. 499, Ex. 5, exponents of ar, z-1. P. 515, line 8, ^. P. 533, Ex. 1, .^±^.
\'
A CODRSE
OP
ELEMENTAHY MATHEMATICS,
ETC., ETC.
The preparation necessary for the profitable study of the following course of Mathematics is — a knowledge of common Arithmetic, and some acquaintance with the principles of Geometry, as taught in Euclid's Elements. A student ignorant of these initiatory, but most important departments of elementary science, would scarcely seek his first lessons therein from a book such as this. The Elements of Euclid is a work by itself ; universally known and esteemed, and everywhere to be easily procured : — to transfer its pages to the present performance, could be of no possible advantage to the learner. And the same may be said of common Arithmetic : — both this and Euclid are more conveniently studied from the ordinary manuals in popular use. We shall therefore commence the volume now in the hands of the reader, with a treatise on Algebra — the indispensable foundation of the entire fabric of modern analytical science.
I. ALGEBRA.
1. Preliminary Notions. — Algebra may be regarded simply as an extension of the principles of Arithmetic. In the latter science the symbols of quantity, to which its rules and operations are applied, are limited to the nine digits or figures 1, 2, 3, 4, 5, 6, 7, 8, 9, together with the cypher or zero, 0. And not only is the notation of Arithmetic limited to these ten symbols, but each symbol is employed by every computer in the same sense : — the character or symbol 4, for instance, stands for/owr, always ; 6 for six; 8 for eight, and so on : the symbols of Arithmetic are thus fixed in meaning, as well as limited in number.
It is otherwise in Algebra: in this science the symbols of quantity comprehend not only the figures of arithmetic, but also the letters of the alphabet : — the figures being, as in arithmetic, of invariable signification, but the letters admitting of arbitrary interpretation. It is this latter circumstance — namely, the possession of a set of symbols which we may employ to represent anything we please — that gives to Algebra its pecu- liarity and its power. In Arithmetic, known quantities only can bo denoted by symbols : in Algebra a quantity altogether unknown, in value, at the outset of an inquiry, may be represented — an alphabetical letter serving this pui-pose, — and then the rules of the science, to be hereafter developed, will enable us ultimately to interpret its meaning, consistently
-84592
2 DEFINITIONS— SYMBOLS OF QUANTITY — SIGNS OF OPERATION.
with the conditions which connect it, in that inquiry, with the known quantities concerned,
2. Definitions— Symbols of Quantity—Signs of Opera- tion.— As noticed above, the symbols by which the quantities operated upon in algebra are represented, are the figures of ordinary arithmetic, and the letters of the alphabet : the marks or signs by which these opera- tions are indicated, are called signs of operation : the principal of these are the following : —
+ , phis, the sign of addition, implying that the quantity to which it is prefixed is to be added.
— , minus, the sign of subtraction, denoting that the quantity to which it is prefixed is to be subtracted.
Thus 5 + 2, which is read 5 plus S, signifies that 2 is to be added to
5 ; and 5—2, which is read 5 minus 2, indicates that 2 is to be subtracted from 5. In like manner a-\-b, or a plus b, implies that b is to be added to a, that is, that the quantity represented by b is to be added to that represented by a. And a—b, or a minus b, implies that b is to be sub- tracted from a. Of course we cannot actually perform the addition and subtraction operations thus indicated, till we know what numbers or quantities a and b stand for.
It may be remarked here, that although the letters a, b, &c. are but the representatives of quantities or numerical values, yet, for brevity of expression, we refer to them as the quantities themselves.
The crooked mark 'v. placed between two quantities denotes the dif- ference between those quantities : thus a~6 means the difference between a and b, whether that difference be the result of subtracting b from a, or a from b.
X , the sign of multiplication, when placed between two quantities, implies that those quantities are to be multiplied together : thus 4x6, or 6 X 4 means that 4 and 6 are to be multiplied together, and axb,or bxa, implies in like manner the product of a and b.
Instead of the sign x , a dot placed between the factors is often used for the sign of multiplication: thus 4.6, or 6.4, and a.b, or b.a, each implies the product of the quantities between which the dot is placed. It must be observed, however, that the dot should range with the lower part of the figures or letters, and not with the upper part, to avoid con- founding it with the decimal point, as, in the case of figures, might otherwise happen : thus 6.4 means 24, but 6'4 means 6 and 4 tenths.
In the case of letters however, the dot is usually dispensed with alto- gether, and the factors simply written side by side, without any inter- vening sign at all : thus, ab, ex, bxy, abxz, &c. mean the same as a x 6. cxx, bxxxy, axbxxxz; or as a.b, c.x, b.x.y, a.h.x.z, &c. This suppression of the intervening sign of multiplication between the fac- tors is not allowable when those factors are numbers, as is obvious : if
6 X 4, or 6.4 were written 64, sixty-four would be implied, and not 24, as intended. But when a single numerical factor enters with the letters, then the multiplying sign may be omitted, since no ambiguity can arise : thus, 6xax6 or Q.a.b, may be more conveniently written 6a6, which means 6 times the product of a and b, or as it is' more briefly read, 6 times a, b. It is proper, as here, always to place the numerical factor first, and the literal factors afterwards ; and also to arrange these latter in the order in which they succeed each other in the alphabet. The numerical factor, thus placed first, is called the coefficient of the quantity
DEFINITIONS SYMBOLS OF QUANTITY — SIGNS OF OPERATION. 8
multiplied by it: thus 6 is the coefficient of ah in 6afe, and 15 is the coefficient of xyz in \bxyz.
-r, the sign of division, when placed before a quantity, supplies the place of the words *< divided by," so that 8-~2 means 8 divided by 2, 12-r3 means 12 divided by 3, a-^-h means a divided by h, and so on; but, as in common arithmetic, division is more frequently indicated by writing the dividend above and the divisor below a horizontal bar of separation, thus: —
•7- is the same as a-^-h, and ~ is the same as 3ajy-r2a6. 0 zoo
8. The four signs now explained —indicating the four fundamental operations both of arithmetic and algebra — are, of course, those of most frequent occurrence in calculation. Algebraists, however, economize their signs of operation as much as possible, and never introduce them need- lessly. This has been already exemplified in the case of multiplication : the absence of sign between letters, placed side by side, as much implies the multiplication together of the numbers those letters represent, as if each were separated from the others by an oblique cross, or a dot. In like manner, when a row of additive and subtractive quantities are con- nected together by the proper signs, if the Jirst of these quantities be additive, or plus, the sign + is suppressed as superfluous; thus : a—b-\- c+d — 4 is the same as -f «— fc + c+(?— 4, and implies that a, c, and d are additive ; or, as they are more frequently called, positive quantities, and that 6, and 4, are subtractive, or negative quantities. If the letters a. 6, c, d stood respectively for 2, 4, 3, 8, the interpretation of the expression just written would be 5.
4. The term coefficient has been already defined : it is the numerical multiplier of the algebraic quantity to which it is prefixed : when this numerical multiplier is simply 1, it is not inserted : it is superfluous to introduce unit-factors; a-|-5, is as well understood to be once a plus once fc, as la + 16; but if the question were asked — What is the coefficient of a or of 6 in the expression «+ 6? the answer would be, not nothing y but 1.
5. =, equal to, is the sign of equality: it implies that what is written on one side of it is equal to what is written on the other, thus : —
7+4=11, 7-4=3, 7-4+1=4, Sx-\-2x-x=ix, &c.
6. Any quantity — of how many letters soever it may consist — is called a simple quantity, or a quantity of but one term, provided it be not sepa- rated into distinct parts by the interposition of a plus or minus sign ; thus, each of the following is a simple quantity, or a quantity of but one term : —
_ _ ^ _ 2ax 14 oaoa;, 7aocy, -^j-, - — , &c.
7. Each of the following, however, is a compound quantity : the first consists of two terms, the second of three terms, and the third of four terms : —
4 2a+8&, 6a— 26+c, 5a5+2cc?— 3m+-.
8. We shall now add a few exercises by which the learner may satisfy himself as to whether he correctly understands what has already been explained or not.
4 ADDITION OF ALGEBRA.
Exercises on the Definitions. — In the following exercises we shall suppose a=4:, b^S, c=6, and d=7.
Find the values in numbers of the following expressions : —
Expression. Interpretation. Vakie.
Baj-2b-c 12+6-5 13.
5b-Za-\-2d 15-12+14 17.
4d^2c-6a+l 28+10-24+1 16.
Sc-d-Zb+ia 40-7-9+16 40.
2a6+5ccZ-16 24+175-16 183.
Consequently 3a^ + 26— c= 13, 56 — 3<i + 2(^=17, 4£Z + 2c— 6^ + 1=15, 8c -d — 36 + 4^=40, and 2a& + 5crf — 16=183 ; and the values of the fol- lowing expressions are to be obtained in a similar manner : —
(1) 26+3a-2d
(2) 48-Ad+dd.
(3) 4ac-\-2cd—db.
(4) Sabc+ibcd-t
(5) ~+bd-ia.
(7)
ab—c 2a ~d 6+c* Scd—idb—l ah
2a6+4 c+d' <^ For the proper values, see answers at the end.
9. Addition of Algebra. — In Arithmetic, addition means the collection into one sum of a set of quantities, all of which are additive or positive : in Algebra the term is extended to the finding the aggregate or balance of a set of quantities, some of which only may be positive, and the others negative ; the result of such addition being plus or minus according as the sum of the positive, or the sum of the negative terms, preponderate. The sum of a set of algebraic quantities — in the absence of all interpretation of the symbols — can be exhibited only when the quantities are all alike; that is, when, so far as the letters are concerned, they do not differ from one another. That a set of like quantities may be added together, without our requiring to know what those quantities are, is plain: thus, it is clear that 2a + 3a + 5a=10fl^, whatever a may stand for, and that Qab + Sab—4:ab=6ab, whatever ab may stand for. If how- ever the quantities are not all like quantities, then to incorporate the entire set into a single term as here, would be impossible. For instance, if we had the set of quantities 4aa; + 2aa;— 3aa; + 26, all we could do would be to actually collect the first three into one term, and then to annex to the sum the fourth term 26 with its proper sign ; we should thus say that 4:ax-\-^ax—Sax + 2b—Sax-{-^b, an expression which we cannot further reduce or simplify till we know something about the values of the letters.
10. Addition therefore divides itself into two cases : 1. When the quantities to be added are all like quantities, and 2. When they are not all like quantities.
Case I. — When the quantities are all like quantities, that is, when they differ in nothing but in their coefficients.
Rule I. Find the sum of the positive coefl&cients. 2. Find the sum of the negative coefficients. 3. Take the difference of these two sums, prefix to it the sign of the greater sum, and then annex the letters common to all the quantities : the correct sum will thus be exhibited.
Note. — When there are two or more columns to be added up, we always commence with the first column on the left, and not, as in arithmetic, with the first on the right, as it is more convenient to write
ADDITION OF ALGEBRA.
the several results, with their signs, from left to right, than from right to left.
The following four examples are worked : the learner should clearly see how the results are obtained, and satisfy himself of their accuracy, before attempting the exercises below.
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(1) |
(2) |
(3) |
(4) |
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Zx |
2ax-\-b |
66a!- 4a |
— 8a6a;+6^ac |
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5x |
Sax+2b |
-26a;+3a |
3a6a;-24ac |
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|
9x |
ax-db |
—56a;— 2a |
— 5abx-\-4:ac |
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|
-X |
5ax-^ib |
12hx-Qa |
— 2a6a;+3ac |
|
|
7x ISx |
iax-5b |
86a;+9a |
Aabx-{-ac |
|
|
Sum i |
ll^ax-b |
196a: ' |
—Sabx+12ac |
|
|
ExEECiS |
3ES TO BE WOEKED. |
|||
|
(1) |
(2) |
(3) |
(4) |
|
|
5a |
3by+ia |
7aa7-j-26y— 3c3 * |
2lmx-7ny-^Z^. |
|
|
-la |
2by-Za |
2ax-Zly-\-icz |
5iMx-\-2mj^Q~. a |
|
|
2a |
- by-\-9a |
Zlax-hy+hcz |
||
|
-^mx—Sny—Q-. |
||||
|
9a |
—^hy-\- a |
'l\ax-{-6hy—2cz |
||
|
15a |
Shy -2a |
—ax-\-Shy — 7cz |
ISlmx—Zny—^-r. |
|
(5) Add the following expressions : laxy—2>h, —Aaxy—2b, 8axy-\-Qb.
(6) 5aa;+l, 3aa;-2, 6aa:+4, -aa;+3, —7ax-5, 4ax+d. (7) 8cz+2x-4, — Zcz—7x, 5cz—ix-{-l, _9c2+3a:-8, 6a;+2, —7. (8) Saxy-^2bz-4.c, —6axy—Bbz-{- 7c, liaxy+5bz~Zc, 2axy—bz—Qc, —iliaxy—bz-12c^ —ldaxy+^bz-\-5c, axy-\-ibz+ 13c.
11. Case II. — When the quantities are not all like quantities.
EuLE. — Collect the like quantities from the several expressions, and add them together : to the sura connect, with their proper signs, those of the quantities which have no like.
Note. — Athough it is of no consequence which set of like quantities be added first, yet the custom is to commence with the quantity at the top of the Jirst column on the left, and to put down, under that column, the sum of all the quantities like it; then to collect the quantities like that at the top of the second column, and so on, as in the following examples: —
(1)
2x—7y-\- iz
Sz + 2x— y
— 2y4-52+ a
Ax-Zz-\- 7y
%x-Zy+14z-\-a
(2)
7ax—2by-{-z
36y+92 —ax
— Qz -\-2ax+by
6by-Bax-\-6
(3) 4:xyz—dxy+ 2yz 6xy -j-Syz — 7ccyz —9ax +xyz'^ll —iyz -\-7xy— 8
5ax->r7hy-\-4z-\-Q —2xyz+9xy-\- Qyz—dax+d
12. — ExEECisEs TO be WORKED. — In the following examples the several expressions may be taken as they are, and placed one under another as above ; or the arrangement of the terms may be changed, so that when placed in vertical columns, the like terms may stand one under
6 SUBTRACTION OF ALGEBRA.
another, as in Case I. ; or lastly, without transcribing, and writing them one under another, we may pick out the like quantities from the ex- pressions as they stand, and write down the sum of each set at once, afterwards connecting, with their proper signs, the unlike quantities.
(4) 4.ax-2y-{-7y dy-8+ax, i2-Bax+p. (6) 6ar-8y+5a, 4y+2x, 20-6a?+7y, 3a-l-6. (6) abx-{-2, 6ay-7-^2abx, 14:— ay, 6ay—9. (7) 8mx—Sny, 5az-^27nx,
Iny-azy 2Z-imx+Qny, ias-B. (8) 2--7-+3, 9^-8, 7?+5, 3aa;-?+2, 8^+
y X X o ox
9-+1. (9) 7dbx—Qaby-^as, 4az—Zay+dbx, ^ax-^7dbx—9aby, 2dbz^6aX'-aby, —
12ay—4az-\-x, 2dbz—5ax—dbz.
13. Subtraction of Algebra. —The operations of Arithmetic are all performed with positive numbers or quantities. In Algebra, quantities both positive and negative equally enter into our computations : we must know therefore how the operation called subtraction is to be per- formed, whether the quantities operated upon be positive or negative.
To subtract a quantity is to take it away from some other quantity : this we can actually do, provided we can split this other quantity into two parts — one of which shall be equal to the quantity to be taken away : the other part is, of course, the remainder.
This simple truth suggests the rule for algebraic subtraction, thus : 1. Suppose we have to subtract 5 from 12 : instead of 13 we may write its equivalent 7 + 5, so that actually taking the 5 away, the remainder is 7.
2. Suppose we have to subtract —5 from 12 : then, instead of 12, writing its equivalent 17—5, and then taking the —5 away, the remainder is 17.
3. Suppose we have to subtract 5 from —12: then, instead of —12, writing its equivalent —17 + 5, and then taking the 5 away, the remainder is -17.
4. Lastly, suppose we have to subtract —5 from —12: then, instead of —12, writing its equivalent —7—5, and then taking the —5 away, the remainder is — 7.
Having thus taken all the possible varieties, as to the signs, of the two numbers — which numbers have, of course, been taken at random — we safely infer the truth of the following results of subtraction : —
From 12 12 —12 -12
Subtract 6 — 5 5 —6
There remains 7 17 —17 — 7
and we moreover see that the very same results would have been obtained if we had changed the sign of each number to be subtracted, and had then added.*
* If we use general symbols instead of figures, the following results, namely, From a a —a —a
Subtract 6 —6 b — &
Remainder a— 6 a+6 —a— 6 — a+6
are shown to be true, by substituting for a the following equivalents, namely : — a=a— 6+6, a^a-jr^—bf — a=— a— 6+6, — a= — a+6— 6.
BRACKETS — VINCULA. 7
We conclude therefore that algebraic subtraction may always be con- verted into algebraic addition, by simply changing the sign of the quantity to be subtracted. Hence the following rule.
14. EuLE. — Conceive the sign of every quantity that is to be sub- tracted, to be changed : and then, with this supposed change of signs, proceed as if the operation were that of addition instead of subtraction : the result will be the remainder.
(1) (2) (3)
From 7aa;-i- %y 5xy— 8az+2 —Saxy-]- hz—2x
Subtract . Zax— 5by _ Sicy-f 2az— 1 5axy-\-7bz-Sx — 6
Remainder iax-\-liby 8xy-~10az+Z ^8axy~6hz-\- x+Q
Note. — It must be carefully borne in mind that when terms occur that have no like, those in the upper row must be annexed to the remainder with their proper signs, and those in the lower row with their signs changed.
(1) From iax—2by-\- 4 subtract 2ax—6hy—Z.
(2) From 2abx-\-Zay—2z subtract 5abx—Say+2z.
(3) From —7xyz-{-5xy-\-Q suhtract —2xyz — Sxy.
(4) From —Zmxy — 6nyz+2asyihtT2iCt27nxy-\-nyz+S. (5)' From Saz—Aby-{-Sx subtract —6x-{-2by—7az-[-l.
(6) From 2^ax-Sby-{-17 subtract 2d-Uiby-lx-\-2z,
(7) From 6^py—9az—m subtract 17az—lpy-{-7n.
(8) From —5ca;s— 7ey-f^2-f a subtract 2<;^—3A;2-fca;«—6-f-4.
15. Brackets — Vincula.— The signs of operation hitherto em- ployed have been prefixed to simple quantities only : when we intend them to apply to compound quantities — or to quantities consisting of two or more simple terms — it is necessary to inclose the terms within hraclcets, such
as ( ), or { }, or [ ] ; or else to cover all with a mnculum, or bar ,
to imply that the several terms thus tied together are to be treated as one whole.
For instance, by writing or tying together the terms in this way, we may express the subtractive operations above thus : —
7ax+ 9hy— {Sax — 5hy) = iax-\- 146y, 5xy—8az+ 2 — {—Sxy -{-2az-'l} =8xy— 10a2-f 3, —daxy-\-bz-2x—[5axy+7lz—dx~-Q]'=—8axy^6bz-\-x-\-6.
16. The minus sign before each of these bracketed quantities implies that the sign of every inclosed term is to be changed on the removal of the brackets : when a bracketed quantity is connected to other quantities by a plus sign, the removal of the brackets leaves the signs of the quantities undisturbed. The following instances of the management of bracketed quantities will be easily understood : —
(a+&)-(a-6)=a-f6-a+5 = 26, a-{b-c)-{b-{-c)=a-b-{-c-b-c=a-2b. a-{-{a—b)—{c—(a—c)}=a-\-a—b—c-{-{a—c) = Sa—b—2c. 2ax—{ax—{2y—dax)}=2ax—ax+2y—Sax=2y—2ax.
17. A multiplier, or coefiicient, placed before a bracketed quantity, implies that the compound whole — or, which is the same — that each individual term is to be multiplied by that coefficient, thus, 4= {a-{-b)= 4a-t-46, 6{a-b)=6a-6b, 3(4a-6 + S)-6^a-Ht-~8)=12a-36-f 6--6a- 66i-48=6a— 9i + 54.
8 SIMPLE EQUATIONS.
18. The equivalence of the following expressions the learner is left to prove for himself —
(1) 4{3a-(6-a)}=4(4a-6). (2) 2{2x-{x-^y)-l}^2(x-y-l). (3) 5{a-^x- 2{x-a)]=5{2a-x). (i) Z{{x-4:)-2{dx-{-2)-5{l-x)\ =-S9. (5) 6{{a-2x)- . d{Ax-2a)}-Z{{ix+a)-{9x-a)}=d6a-69x.
19. Simple Squations. — An equation is merely a declaration, expressed in the characters of algehra, combined or not, with those of arithmetic, that two quantities are equal. Thus : if a; be such a number that a;— 3 is equal to 8, then the algebraic statement of this equality, namely, a;— 3=8 is an equation. It is plain that to satisfy Ihis equation, a must represent the number 11. In like manner, if x be such a number that 4a;— 3 is equal to 2^7 + 7, we shall have the equation 4a?— 3 = 2a; + 7. The finding the value of the unknown quantity — in this case x — that is, the discovering what the unknown quantity really stands for, is called the solution of the equation.
20. By help of the few principles established in the preceding pages, the more easy kinds of simple equations may be readily solved. It will be only necessary to observe the following particulars.
1. Transposition. — Any term may be taken from one side of an equation and carried over to the other side, provided that, when thus transposed, its sign be changed. For by thus transposing a term, the balance, or equality of the two sides of the equation, remains undisturbed: the result is still an equation. Suppose, for instance, the term 'Sah occurs on one side of an equation, and that we wish to remove it from that side without disturbing the equality of the two sides. All we have to do is to add — 3a& to both sides, which addition, it is plain, removes the 3ah from the one side, and transfers it, with changed sign, to the other. In like manner, if the term —5a;, standing on either side of the sign of equality, is to be removed, we have only to add 5a? to both sides, which addition merely transposes the —5a; from one side to the other, on which other it re-appears with changed sign. And it is plain that if the two sides were equal before the transposition, they must be equal afterwards.
For example, take the equation above; namely, 4a?— 3=2a; + 7. By transposing the 3, we have 4a;=2;i? + 7 + 3 : and by transposing the 2ar, 4a; — 2a;=7-|-3, that is, 2a?=10 : consequently a;= 5, which is the solution of the proposed equation 4a?— 3=2a? + 7, as it is easy to see, for 4.x, that is, 4 times 5, is 20 ; and 2a?, that is, twice 5, is 10 : each side of the equation is therefore 17.
2. Clearing Fractions. — Whenever a fractional term occurs in an equation, we may free the equation from the fraction by simply multi- plying both sides by the denominator of that fraction. And in this way may fractions be removed one after another : the final result still being
2a; an equation. For example : if the equation be — +4a;=7, by multiplying
o
each side by 3, we convert it into the equation 2a; + 12a;=21, that is,
21 3 14a;=21, so that x———-, or 1^. 14 2
21. In fact, the principles brought into operation in the solution of a
simple equation are all justified by the following axiom ; namely, that
whether we equally increase or equally diminish, equally multiply or
TO SOLVE A SIMPLE EQUATION, ETC. 9
equally divide the two sides of an equation, the result is still an equation; or, generally — whatever operation we perform on one side of an equation, if we perform the same on the other side, the result must be an equa- tion.
22. To solve a Simple Equation containing only one
unknown Quantity. — The symbolical representation of an unknown quantity is usually the letter x, or y, or z: the earlier letters of the alphabet are employed, almost exclusively, to represent known quantities. A question may be proposed in which it is declared that certain quantities are known, although their actual values may not be specified : in such circumstances, we should represent the known quantities by a, b, or c, etc. The rule for solving a simple equation, in which all the quantities but one are known, is as follows : —
Edle I. If a fraction occur in the equation, clear it away by multi- plying both sides of the equation by the denominator.
2. By transposition, bring all the terms containing the unknown quantity to one side of the equation, placing the known terms on the other, so as to get an equation in which the quantities on one side are all unknown, and those on the other, all known.
3. Collect the quantities on each side into a single term : there will then be a single unknown quantify on one side of the equation, and a single known quantity on the other.
4. If the X (or the y, or tke z, as the case may be) in this unknown quantity, have a co-efficient, other than unity, divide each side by that coefficient : then the x will stand alone on one side of the equation, and a known quantity, which is its interpretation, on the other.
23. Examples. — (1) Find the value of a; in the equation 9^—5 = 3ar + 19. As there are here no fractions to be cleared, we commence with the second precept of the rule, and transpose: we thus get 9a;— 3a;=19 + 5. Collecting the terms, 6x=24. Dividing by 6, a:=4 : hence the quantity X, at first unknown, is found to be 4: we see that 9 times 4 minus 5 is equal to 3 times 4 plus 19, as the equation affirms.
(2) Find the value of x in 4— — -+2a;=ll. In order to clear the
fraction, we multiply by 6, and thus get 24— 5a?+12.r=66 This, by transposition, becomes — 5a;-|-12ic=66— 24
which, collecting the terms, is 7ii?=42
60 that, dividing by the co-efficient 7, we have finally x=Q.
(3) |-|+5=x-5
Trans., ^-|=a:-10
2x Mult, by 2, x—-^ =2a;— 20 o
Mult, by 3, Zx-2x=:Qx—Q0
Trans., Zx—2x—Qx= — QQ
Collecting, — 5a;=— 60
or Trans., 60=5a;
-r 5, 12=.x.
<'> —+6-4=^
Mult, by 2, a;-l+|-|=0
Zx ^ by 3, Zx-Z-\-x- -=0
by 2, Qx—&-{-2x-~Zx=0 Trans., 5x—Q, therefore a;=l^.
(5) ?^^+6=3(.:-3)
Mult, by 2, 3(a;--2)+12=6(x-3) that is, 3:r-6+12=6x-18 Trans., 3a;-6.r=-18+6-12
that is, — 3.^;=-24
or Trans., 2i=^Bx -i-3, 8=;^.
10 SOLUTION OF QUESTIONS BY SIMPLE EQUATIONS, ETC.
24. Examples for Exercise.
(1) 8-3a;-|-12=30-5^+4.
(2) 4a;+3=3(a;+4).
(3) 2(ar-3)=3(6-a;)+2.
(6) ^+|-f=to-17.
(7) ^_?+^+l=o. ^'^ 3 4^5^2
(8) aa;— 6=cx4-»>i.
(9) (a+&)ar=6;»+4.
ao) 5<?^)=iizS!.
25. Solution of Questions by Simple Equations with
one unknown. — Questions implying but one unknown quantity are solved by Algebra by representing that unknown quantity by a letter, as X, by then expressing the conditions of the question in the form of an equation, and solving that equation as in the foregoing examples.
Note. — To avoid the frequent repetition of the word therefore, in the steps of an algebraic operation, the symbol .'. is used to stand for it : this symbol reversed (namely '.•) is frequently used for the word because.
Examples. — (1) What number is that whose third part exceeds its fourth part by 8 ?
Let X be the number, then by the conditions of the question,
f-^=8.-.Mult. by3, a;-?^=:24. o 4 4
And, mult, by 4, 4a;— 3a;=96, that is, ic=96. The third part of this is 32, and the fourth part 24.
(2) Divide the number 48 into two parts, such that one part may be
three-fourths of the other.
^x Let X be one part, then — is the other ; so that
«+— =48. Mult, by 4, 4a;+3a;=192 .*. 7a:=192 .-. a;=27?.
Hence one part is 27f , and the other is therefore 48—27^=20*-.
(3) If A can finish a piece of work in 8 days, and B can do it in 10 days, in what time will they finish it both working together?
Suppose in x days. Now the part of the wliole, done in 1 day, is, by
11 9 9^ •
the question, 0+77., that is, — : hence, in x days, -- is done: but
this is 1 whole.
.*. T7:=l. Mult, by 40, 9a:=40 .-. xz=i^ days.
(4) A vessel can be filled by four taps, running separately, in 2 hours, 3 hours, 4 hours, and 5 hours : in what time will it be filled if all run together ?
Let X be the number of hours. The part of the whole filled in 1 hour
11 X
is i -1-1.+^ -j-^ or J J. Consequently the part filled in x hours is — -•
But this is 1 whole.
.-. g^=l ••• 77a;=60 .-. x=Y^ hours.
Hence the vessel will be filled in the -f ^ part of an hour.
(5) A child being asked the time by the clock, answered that the hands were both together between 5 and 6 : required the exact time.
SOLUTION OF QUESTIONS BY SIMPLE EQUATIONS, ETC. 11
Suppose the hands are at a? minute-spaces past the 5, then, while the short hand has moved over these x minute-spaces, the long hand has moved up to it from the 12, and therefore over 25+^ such spaces. But the long hand moves 12 times as fast as the short hand,
.-. 12x=254-a; .'. lla:=25 .-. x=2^ .-. 25+a;=27A,
the number of minutes past 5 o'clock.
(6) What number is that of which the fifth part exceeds the sixth part by 7?
(7) What number is that which when increased by one-half and two- fifths of itself, the sum may be 76 ?
(8) Divide the number 30 into two parts, such that one part may exceed the other by 13.
(9) Divide £100 among Ay B, and C, so that A may have £20 more than ^, and ^ £10 more than C.
(10) From two towns, 144 miles apart, two persons set out at the same time to meet each other; one travels 16 miles a day, and the other 20: in how many days will they meet ?
(11) A can finish a piece of work in 15 days, which B can do in 12 days : in what time can they finish it, both working together ?
(12) A railway-train leaves London at 12 o'clock to run to York, a dis- tance of 200 miles ; another leaves York at the same time for London : the former train goes at the rate of 25 miles an hour, and the latter, at the rate of 35 miles an hour. At what o'clock will they pass each Other?
(13) A detachment of soldiers march from a certain place at the rate of 2| miles an hour, and two hours afterwards another detachment, marching at the rate of 3^ miles an hour, is sent to overtake them : how far must they march to do so ?
(14) A garrison of 1000 men is victualled for 25 days ; but after 9 days 250 men are withdrawn: how long will the provisions last those who remain ?
(15) Out of a cask of wine, which had lost ^ of the whole by leakage, 30 gallons were drawn : it was then found to be half full : how much did it hold ?
(16) How much tea at 4s. 6cZ. per lb., must be mixed with 50 lb. at 6s. per lb., so that the mixture may be worth 5s. per lb. ?
(17) A vessel holding 120 gallons is partly filled by a spout which delivers 14 gallons in a minute; this is then turned off, and from a second spout, delivering 9 gallons in a minute, the filling of the vessel is completed : the whole time occupied is 10 minutes : how long did each spout run ?
(18) A market woman had a certain number of apples at 2 a penny, and as many more at 3 a penny ; but resolved to sell the whole at the rate of 5 for twopence; and, contrary to her expectation, found she lost M, How many apples had she ?
(19) A person pays a bill of £2 16s. M. with half-crowns and shillings; the number of pieces was 40 : how many were there of each ?
(20) A person has three debtors, A, B, C, whose debts he only so far remembers that A's and B's together amount to £60, A's and C's to £80, and J5's and Cs to £92 ; required the debt of each ?
(Additional examples in simple equ^ions will be given further on).
12 DEFINITIONS.
26. Definitions. — Powers. The product of any number of equal factors is called a power of the factor thus repeated. It is the second power when produced from two equal factors, the third power when pro- duced from tliree equal factors, and so on. For example, since 2 x 2=4, 4 is the second power or the square of 2, and since 2x2x2=8, 8 is the third power or cube of 2. In like manner, 2x2x2x2 or 16 is the fourth power of 2, 32 is the fifth power of 2, and so on. Similarly with letters, aa is the square or second power of a, aaa is the cube or third power, aaaa is the fourth power of a, and so on.
27. Exponents or Indices. The inconvenience of thus repeating the equal factors in expressing powers is avoided by writing the factor only once, and marking the number of factors — that is, the power intended — by a small figure placed over the right-hand corner : thus, 2"^=4, 2'^=8, 2^=16, 2^=32 ; «', a\ a\ a\ &g,
These small corner figures are called eccponents, or indices. Conformably to this notation, the ninth power of x is x^.
28. Roots. The number or quantity which, by repeated multiplication by itself, produces a power, is called a root of that power ; thus, 2 is the square root of 4, 3 is the cube root of 27, 2 is the fourth root of 16, &c. And generally any quantity of which the square, cube, fourth power, &c., produces another quantity, is the square root, cube root, fourth root, &c., of that other quantity. The notation for roots is similar to that for powers : thus, 92, 273, 16^ &c., denote respectively the square root of 9 or 3, the cube root of 27, which is also 3, the fourth root of 16, which is 2, &c. But there is another way of indicating roots, namely, by means of the radical sign, \/, thus : 9% 27^, 16^, aK a^^ &c., are the same ass/ 9, v^27, v- 16, \/a, s/a; &c., where for the square root, the small index-figure, 2, is suppressed.
29. To indicate powers or roots of compound quantities, the several simple terms must be inclosed in brackets, or united by a vinculum : thus the square of the expression axi^—bx'--\-c would be indicated in one or other of the following ways, namely : —
{ax'-hx^-\-cY, ov[a3?-lx^-^cf, or {ax'-lx'-VcW or '^-hs^-^c and the square root by —
{ax^—hx^-^-c)^, or [aa;'— &a;^-l-c]2> or {aa^—hx^-^-c}^, or aa?—hx^-\-c^ Or ^{ax^—lx^-\-c), .y\ac(?—'bx^-\-c], »/ {ao^—hx^-{-c}, .yax^—hx^+c
30. The operations of addition and subtraction, hitherto applied to simple quantities only, may be easily extended to more complicated ex- pressions, such as these. So long as the expressions are like, they may be just as readily incorporated, whether they are simple or compound: it has been sufficiently seen that it is with the coefficients only of these like quantities that we deal. Any multiplier prefixed to an expression is the coefficient of that expression ; and this term coefficient is extended to mean the prefixed multiplier, whether it be numeral or literal. Thus in 2>{x'-^y+z), a{x'—':iy + z), {a + h){x''-'ily-\-z), (2a-5)(a7-— 3 '^y-\-z), the multipliers 3, a, a-\-b, 2a— 5, are the respective a coefficients of (o;^ — 'iy-\-z). If the above four quantities are to «+5
be added together, then proceeding with these coefficients, as in ^°^~^ the margin, we should find the sum to be (4a-i-6— 2)(a;'^— 4aj_j_2 2.V+^).
MULTIPLICATTON. 13
Examples. — Add together the following expressions :—
1. 7(^+2/), {a+h){x^y), {2a-h){x-^y), {2,h-^){x^-y).
2. 4— 2v'(a;— a), 5^/a;— a+1, Z{x—a)^—Q, 2—[x—af^.
3. ^{x-y)-^{x-^ry), a^{x^y)+6^{x-y), {Za-{-<2)V{x-y)-'l{x-\-y)'i.
4. Subtract Zz+4:x'^-2y'^-\-h from 2x^-6y^+^z-Ji-a.
5. Subtract ^{x—y) — {a—h)xy-^Qz from 3a(a;— j^)— 4a:y+22.
6. Subtract —^x-\-y)-\-^x^—y'^)-{h+c)zivomQ{x-\-y)—a{x'^—y'^) — (p-\-c)z.
7. Subtract — 6av'y+35v'a;+a+d from 4aa;^— 5%^-^a— 5.
31. MultiplicatioxiM — In writing down the product of two algebraic factors three things must be attended to : first the sign, then the coefficient^ and, lastly, the letters.
If the signs of the factors be like, that is, both plus or both minus, the sign of the product is plus.
If the signs of the factors are unlike, the sign of the product is minus. This may be shown as follows : —
Take any two numbers at random for factors — say 8 and 3 — then all possible varieties, as respects the signs of these factors, will be compre- hended in the four following, —
8 - 8 8 - 8
3 3 __ 3 - 3
24 -24 -24 24
The first, being the case of common arithmetic, requires no remark. In the second case the product must be— 24, because multiplication by 3 means repeating the thing multiplied three times. To discover the cor- rect product in the third case, we may proceed thus : add 4 to the multi- plier; then it is plain, that by operating with this altered multiplier, the product we shall get will be 8 x 4, or 3Q too great.
The altered multiplier will be — 3-t-4, that is 1 ; and 8x 1=8. Sub- tracting, then, the 32, we have 8— 32 =— 24 for the correct product.
In like manner in the fourth case, the increased multiplier being — 3 + 4 = 1, and the product by it — 8x 1 = — 8, we have by subtracting —8 x4 or — 32, 24 for the correct product*
I. When the factors are simple quantities. — Rule. 1. If the signs of
* The proof is the same when letters are used instead of numbers ; thus : — 6 - b h - h
a a —a — a
db —ah —ah • ah
The first and second results are obviously true. Taking the third, add a-\-l to the multiplier, then to correct the product, we must subtract from it a+1 times h, that is, a times 6 and onceb, which is ab-\-h. Now the product from the new multiplier is once h ovb ; and subtracting ab-\-h, the result is —ah. Increasing the last multiplier in the eame way, the product is —h, and subtracting a+1 times —b, that is, —ab—h from this, the result is ah. To explain the meaning of an isolated negative quantity, the learner may be reminded that -\- and — stand in opposite relation to each other. If -f^4 represent a gain, then — £4 represents a loss. If +4" express the elevation of an object above the horizon, then — 4® expresses its depression below the horizon ; if -\-a denote time after any epoch, then —a denotes the same time before that epoch; and so on.
14 MULTIPLICATION.
the two factors are like, put 4- for the sign of the product ; if they are unlike, put — .
2. After the sign write the product of the coefficients.
3. To this product annex the product of the letters, that is, write the letters in both factors one after another, without any intervening sign.
Note. — The letters of the product are usually arranged in the order in ■which they follow in the alphabet ; but when the same letter recurs, the notation for powers is used ; thus, instead of aaa, we write a■^ instead of i»V\ or which is the same, asx x a;xx, we write a^, the sum of the ex- ponents in the factors being the exponent in the product : thus, generally, a?'»xaj'*=a?'"+", whatever integers m and n may be.
(1) (2) (3) (4)
Multiply 5ax — 8a;'y 7ahx^ — Gab'^xY
by Zab 2ai/ — iax"^ — Aa^xi^^
Product Ua'ix -IQaxY -2%a%a? 24a^iV3/6
(5)9cajyx-2aca;j/-=-18ac%y. {^) -\x^yy,ia3^=i—~<Ls^^y. {1) —abxy^X —
5a26V=5a3JV2/i (8) ^dbx^X-1a'¥x=-bQa*V'a^. (9) -ix^ykx-Zaxly^=12axy. Examples for Exercise.
(1) Za^xXQaHxy. (2) -Ui/xlaVxy^ (3) -baVxy^X-Za^hxy. (4) laxyX la'xfx-lxy. (5) -6a:/x3a6Vx-2j/2x6. (6) -^ax'^yX-WX-'tcyz,
II. When one of the factors is a Compound quantity. — Eule. Multiply each term of the compound quantity by the simple factor, commenciog the operation on the left hand; the several partial products, connected together with their proper signs, will be the complete product.
(1) (2)
Multiply 2ax'^-Zly 6aV-4&a:V-3c3*
by ia^x^ — 4axy
Product 8a^xr'-12a^bx^y -2ia''x'y+16abxY-\-l2acxyz\
Examples for Exercise.
(1) {iabx'-2xy'')xBa%x. (4) {ax+(2ly-[-S)}Xiaxy.
(2) {GaxY-\-lSb^-5)x-2abxz. (5) ^ax'{Aba^-{dax-\-l)}.
(3) {2ax—dby-{-4:C2)XaxX—bc. (6) {Qab'^x—5cy)X—iaxXiby.
III. When both factors are compound quantities. — Rule 1. Multiply every term of the multiplicand by each term of the multiplier. 2. Ar- range the se\er&Y like products one under another, and find their sum by Addition —
(1) (2) (3)
Multiply a-\-h a—b a +6
by a +6 a — 6 a —6
a^4- ah a^— ah a^-\-ah
a&+52 — a&+5' —ab-h^
Product a^+2a64-62 a'^-2ab+b^ a'-b^
MULTIPLICATION. 15
Hence, whatever be a and??, we see that(a+6)^=a^+2a6-f 5^ (a— 6)-= a^^2ab-\-b', and (a-^b) {a — b)=a^—b^, general truths which should be borne in remembrance. They show that the square of the sum of any two quantities is equal to the sum of the squares of the quantities themselves, plus twice their product; that the square of the difference of any two quantities is equal to the sum of the squares of those quanti- ties, minus twice their product; and that the sum of two quantities multiplied by their difference is equal to the difference of their squares, and, consequently, that if the difference of the squares of two quantities be divided by the sum of those quantities, the quotient must be the difference of the quantities ; and if divided by the difference, the quotient must be the sum.
(4) (5) (6)
X — y ix —5 2x —h
a?^xhj-\-xy'^ ^x^-\1x^ 2ax^-\-2hx'^-{-2cx
—x-y—xy^—y^ —10x'^-\-15z —abx^—b^x—bc
x'-f. 8a;3_22r*-fl5a:. 2ax^-{-{2b-ah)x'+{2c-b')x-bc.
The product in Ex. 6 may be written thus: 2aa;^+(2—a)6a;--j-(2c— b-)x—bc.
(7) (8)
^—xy-^-f 12ay — fiab +3
X 4-y ay — lab ■\- 2
i)?-xy-\-xf 12aV- 5a%-|- %ay
x'y-xy^-^f -2ia%y •\-lQa%'^- Qab
2iay —lQab+6
x'+f.
12ay-29a%-|-27a3/+ lOa'62- 1 6ab+6.
When, as in examples (1) and (2) above, the factors multiplied together are all alike, the operation is usually called Involution : the result of the involution is, of course, a power of the quantity thus multiplied into itself. Although special examples in involution are not absolutely neces- sary, we shall here exhibit two for the sake of the inferences to be drawn from them, and which will be made use of hereafter.
(8) Required the cube of a + 6. (9) Required the cube of a—b. {a-\-by=a^-{-2ab +6» {a-by=a''-2ab +5*
a +5 a —b
oH2a26-fa6» a^-2a^b-{-ab^
a^+2ab^-\-b^ — a%+2ab'^-b^
(a+5)»=a3-f3a'J+3a&2+&3 {a-bf==a^-Za%-\-Zab''-b^
These two results may evidently be written as follows ; namely, —
(a+J)'=a='+&='+3(a+6)a&, or a^+{Za'^^-Ub-\-b^b.
(a-by=a^-P-S{a-b)ab, or a^-{Sa^-dab-^b^}b. and in these forms it will be advisable for the learner to recollect them ; the symbols a, 6, may of course be replaced by any quantities whatever.
16 DIVISION.
Examples for Exercise.
(8) {Sx-2){2x+Z){Bax-l)=
(9) {ax"'+lx''){bx"'+ax")=
(10) {kx+2){x'-ix+l)=
(11) {x^—^x){x»-ix-\-l)=
{12) {ax-{-b7/){ax-bi/)= [See page 15.] {IS) {Zax-7byy=: [See page 15.]
(1) Multiply rc2-l-2a5;+a^ by a;2-2aa:+a'
(2) {x'-\-x-'-^l)(x^-l)=
(3) (a;*+aV+a*)(a;2-a2)=
(4) {x+l){x'^-\x+l)=
(5) {Sx-i){2x'^-5x+2)=
(6) {x-y){x'-^x''y+xf+f) = <7) {x'-2x+l){2x^-\x+2)^
32. Division. — As in multiplication, three things are to be attended to : the sign of the quotient, the coefficient of the letters, and the letters themselves. Whether in multiplication or in division, like signs give plus, and unlike signs minus.
I. When dividend and divisor are both simple quantities. — Rule 1. Write the sign of the quotient. 2. Then the quotient of the co-eflBcients. 3. And finally, the quotient of the letters : — these being such that, when joined to the letters of the divisor, they make up those of the dividend.
Note. — Should letters appear in the divisor which are not found also in the dividend, the division by thein cannot of course be actually per- formed ; and, as in arithmetic, after the quotient by the other quantities is obtained, the unemployed factors of the divisor must be written under it, to imply a further division which has not been executed.
(1) Divide — 12d^xY hy Sax-yz. Here the coefficient of the quotient is —4, and to this we are to annex such letters, that when they are united to (multiplied by) those in the divisor, they may make up the letters of the dividend. The factor z in the divisor, not occurring in the dividend, must be reserved : omitting this, we see that the letters wanting in the divisor to make up those in the dividend are a^x^y ; hence the
. —Aa^oj-y . . — ISrtVw^ —^c^a?y 4a Vw
quotient is , that is to say, — ^ — -, — ^=' -, or -,
z oaxyz z z
(2) 18a6Z;V2/-i--9a%2=-l^l-l^. (3) ~14a&W/^ — 2b'anJ=7abxY
8a^c^_8aT£*
6aWc-a!^y^~ bxy^ '
Note. — As these illustrations sufficiently show, when powers of the
same letter enter both dividend and divisor, the quotient, so far as that
letter is concerned, is got by simply subtracting the lower exponent from
the upper : the remainder is the exponent to be given to that letter in the
^4 J5 g6
quotient. In the fourth example, for instance, -^=a^ T5=^^ ~2— ^*-
d/" 0 c
But when the lower exponent exceeds the upper, then only the part of the lower exponent, equal to the upper, is subtracted, and the lower letter, with its exponent thus diminished, is still retained as a divisor; since thus much of the division is still unperformed. In the above
a;'^ 1 w^ 1 example, for instance, -^=— , —5=— J the learner thus perceives that
when only powers of the same letter are concerned, multiplication merely implies the addition of the exponents, and division the subtraction of them.
division. 17
Examples foe Exeecise.
(1) Divide -15aV^» by Sa^'xy'
(2) 18a%^xy-^-6ah^xi/=.
(7)
(3) -27a:x^i^^z^-Zaxf=.
(4) 2al^x^yh-^a'bx'^^ = .
(5) -21a'b*c'a^y-^9a%'c*x^ = .
(6)
14a^hx*y^
(8)
2a%x^s/z \<m>'^x>/z
II. TFAew <^5 dividend is a compound quantity, and the divisor a simple quantity. — Rule. Find the quotient of the divisor and each term of the dividend, as in last case, and connect these several quotients together by their proper signs.
._, Sa^x^v*— 4.i;3y4 , „ „ ^ ,^^ iax-^—Sa^xz^—4xz ^ „ 1
^^^ |^27-^=4aV-2:ry. (2) —^ axz-2a^--.
Examples foe Exeecise.
{1) a*ar^-Zahx^+5ax*^ax^. (2) ?^f-=|^^±l?. (Z)-12ahx^2+%^]/^z-eb!^-^-Zlz,
(4) 3{2:«;-(164-a;)+l}-r5. (5) -2ia''x''y-Baxy-\-6xy-^-Zxi/.
(6) labc^X'r-Wb''cxk
III. When dividend and divisor are both compound quantities. — Rule. 1. Arrange the terms both of dividend and divisor, so that the powers of some letter common to both may follow each other in ascending or descending order.
2. Divide the first term of the dividend by that of the divisor ; the result will be the first term of the quotient. Multiply the whole divisor by this first term, subtract the product from the like terms of the dividend, and to the remainder annex a new term of the dividend, or two terms if necessary; regarding the result as a new dividend.
3. Proceed with the divisor and this dividend as at first, and continue the operation till all the terms in the original dividend have been brought down. If there be a final remainder, it must be written, with the divisor uudemeath, and annexed to the quotient, as in arithmetic.
Note. — When divisor and dividend are seen to have a factor in common, this common factor may be cancelled from both before pro- ceeding with the operation.
(1) Divide ISa;^ — 13^*— 34^^40.1? • by ix^'—lx. Here it is seen that the factor x is common to both quantities : expunging this common factor, therefore, we proceed by the rule as follows, the terms being already ar- ranged according to the descending powers of x: —
Ax-1)\2x^-lZa?-Zix^-^i0x{Zx'-\-2x^-5x-\-j£jj 12a;*-21a;3
%x^-Ux^
-2Qx^-\-iO« -20x^+Z5x
Remainder 5x
18 DIVISION.
(2) Divide 4a''a;+5laV+10;c*— 48a:»^-15a* by 4ax-6x'' + Sa\ Arranging the terms in order, according to the powers of x, the opera- tion will be as follows : —
-6a;2+4ax+3a2)10a:*-48aa;3+51aV+4a3a;-15a*(-2u;H8aa;-5a2 10a:*- Saar'- 6a V
-40a;c3+57aV+ 4a^x -40aa:3^32aV+24a'a:
25a V- 20^3^- 15a« (3) Divide 1 by 1+ar. 25a^x^-20a^x-15a*
1+^)1 il-x+z^~^ l+ar
—X
—x—x^
The division in this third example may be carried on to any extent ; so
that (omitting the remainder) — -=l— 4;+a;'— aj'+a;*— aj^+a?"— a?''4- &c.
But at whatever term of the quotient we may choose to stop the opera- tion, the subsequent remainder, with the divisor underneath, must be connected, with its proper sign, to the quotient.
\ 3? a:*
Thus, — — -=1— aj+a:*— — — , or z=:\—x-\-3?—s?-\--——^ or =1— ar+a;*— a;^+a:*— 1+a; \-\-x \-\-x
— — -, and so on.
Examples for Exercise.
(1) Divide 6a:2+13a:+6 by 3a;+2.
(2) „ 6a;*- 96 by 3a; -6.
(3) „ «^-/ by x-y.
(4) ,, a^—ar^ by a— a;.
(5) Divide 25a;''-a;*-2a:3-8a;2ty5«3_
4a;2.
(6) „ 6a^+9a;2_20a;by3a:2_3^^
(7) „ Ibyl-ar.
(8) „ :^-\-lx'^-\-cx^d by x-a.
and show that the remainder is the same as the dividend when the x in the latter is replaced by a,
33. Note. — In this last example the truth of the property mentioned is supposed to be arrived at by actual division, the remainder from that division being a"^+6a^+ca+ci; the property, however, is perfectly gene- ral, and may be established without going through any division process. It may be enunciated as follows : —
If any algebraic expression with terms containing x, when divided by either x—a^ or a; -fa, leave a remainder free from x, that remainder will always be what the dividend becomes when the x in it is replaced by a, or —a, according as the divisor is «— o, or x-\-a.
For, from the nature of division.
Quotient x Divisor -j- Remainder=Dividend.
ALGEBRAIC FRA.CTIONS. lO
And in the case before us, the Eemainder, being free from a?, must continue unaltered whatever value we put for a: in this equation. Let a be put for o), if the divisor be a— a, ov —a ii it he a-\-a; then the above equation will be
Quotient X 0 + Remainder = Dividend.
But any finite quantity multiplied by nothing produces nothing ; hence the equation last written is simply Remainder = Dividend; that is, when the proposed substitution for x is made in the dividend, the changed ex- pression is the same as the remainder. The following are illustrations of the application of this theorem: — (1) Is ^*— 3;k"^— 5^— 14 divisble by ;»+2? Putting —2 for a?, the dividend becomes 16 — 12 + 10 — 14, which is equal to 0. .-. Remainder =0, .-. the expression is divisible by ^+3. (2) Is x^^^ar^ + ^x—4: divisible by ic— 2 ? Putting 2 for ^ in the dividend, we have 8—8 + 6—4=2, the rem. Hence the proposed expression is not divisible by ^—2 ; the division leaves a remainder 2. The following also are useful inferences : —
Whenever n is odd \ ^"-*" ^^ divisible by x-a, but not by x-]-a ( a;"+a'* ,, x-\-a, but not by x—a
TTTi. • { x^ — a" ,, both a; — aaindx-\-a
Whenever n is even ] " . , ^"'
{ x^-\-a^ ,, neither a;— a nor a;+a
By the principle above, the remainder, in this last case, is 2^".
34. Algebraic Fractions. — Fractions in Algebra are treated ex- actly in the same way as those in Arithmetic: whether the symbols employed be figures or letters the rules of operation are just the same. This the learner will be prepared to expect, because algebra becomes con- verted into arithmetic so soon as the letters are replaced by numbers. The Rules therefore need here be but very briefly recapitulated.
Reduction of a mixed quantity to an improper {or single) fraction. — Rule. Multiply the integral part by the denom. of the fractional part: connect the product, by the proper sign, to the num., and write the denom. underneath: thus —
(l)a+^=^. (2)a+._?5=^^±^?^^. (3) ^^+^=^±^^=.^.
^ ' 6 b ^ ^ ^ y y ^ ' ^ ' ^X s/X ^X
(A\ <^°+^' a:'-2ah-\-h''_{a-lf
ah ah ah
Examples for Exeecise. — Reduce each of the following mixed quan- tities to an improper fraction : —
W-+;- (^)l-T (3)1-^' (4)^%2. (5).»-3.+2+^l
Reduction of an improper fraction to a whole or mixed quantity. — Rule. Divide num. by den. If there be a remainder, annex it, with the denom. underneath, to the quotient, as in all cases of division.
^ ' 6 by y
^ ' 2ax ^2ax ^ 2ax
0 2
5J0 ADDITION AND SUBTRACTION.
Examples for Exercise. — Keduce each of the following to a whole or mixed quantity : —
^^^"T"- ^^^ x-y ' ^^K-\-b- ^^^ d^x • ^^K-^-3x+2'
^^^ 4ax • '^' 4x^-7x '
Iteduction of fractions to a common denominator. — Rule. 1. Multiply each numerator by the product of all the denominators except its own, the several new numerators will thus be obtained. 2. Multiply all the denominators together, the product will be the common denominator.
That the values of the fractions remain unaltered by this change in their forms is obvious ; for, take whichever of the original fractions we may, we see that the change is effected by multiplying its num. and den. by the same thing ; namely, the product of the denominators of all the other fractions. The common denom. found as above is, of course, a com- mon multiple of all the original denominators : if it be not the least common multiple, the changed fractions will not be in their most simple forms. By glancing at the original denominators, the least common multiple of them may often be readily found : — if common factors enter two or more of the denominators, they should be retained in but one : — the repetitions should be struck out; the product of the denominators, thus deprived of common factors, will be the L.C.M. (least common mul- tiple). As to the numerators, we have only to multiply each by the quotient arising from dividing the L.C.M. by the corresponding denomi- nator, so that, as before, the values of the changed fractions remain undis- turbed.
The object of this reduction of fractions to a com. den. is merely to fit them for addition and subtraction, the rules for these operations being as follows.
Addition. — Rule. Reduce the fractions to a common den., and write this com. den. under the sum of the changed numerators.
Subtraction. — Rule. Write the com. den. under the difference of the changed numerators.
(1) Eequired the sum of - — , — , j. Appljring the rule, the new numerators are
2x8a?, 5xl2aa?, cY^^au^^ and the com. den. is 3aafX2d?X4:: hence the changed frac-
16j? Vlahx Qacx^ . , . , .i ^ « • « i? j. • ^i.
tions are z, s> 5> m which we see that 2x is a superfluous factor in the
24a^ 24«j?2 2iaar
num. and den. of each. We might have foreseen this by running the eye along the row of denominators : — there is a repetition of the factor x and of 2 : suppressing, there- fore, one a and one 2, we have 3a,»x2x2=12aa? for the L.C.M. of the denomina- tors. Hence multiplying the numerators in order by 4, 6a, Sax, and adding the results,
8+6a64-3ac/r , ^,
we have for the sum.
12ax
2x4-3 X 1
(2) — — — I — I — . Here the denominators have no com. factor; hence by the rule,
4 6 3
30a;+454-123;+20 42a;+65 . ^^ 60 = -60- ^ *^' '^-
(3) -^ — — I — ^^^. Here 28 is the L.C.M. of the denominators.
^ ' 4 7 28
14a;-21-12a;-16+5-2a; 32 8 ,^ .*. = = — , the sum.
28 28 7'
MULTIPLICATION AND DIVISION.
21
x+y 3^—y^ {x+y){'^—y) v?—y^ x^—y^ x^—tf
Examples for Exercise.
a±^ a-x
(2) %x^
14'
(3) ^+^-^^2=.
(4)
34-2.g 2__
1-0^ \-x~'
(i>) -S 2"^ 1 =•
(6) Prove that -^+-^=-'^ ^.
x-\-y x—y x—y x-\-y
(7) Prove that — ?^=_^ !_.
Multiplication and Division. — For Multiplication. — Rule. 1. Multiply the numerators together: the result will be the num. of the product. 2. Multiply the denominators together : the result will be the den. of the product.
For Division. — Rule. Invert the terms of the divisor ; that is, turn the fraction upside down, and then proceed as for multiplication. The
truth of these rules may be proved thus : — Let -, - be two fractions, which
represent by x, y respectively : then
Cti c
4?=-, and y=^- . '. hx=a, and dy=zc . '. hdxy=:ac
o d •"
ac . , a c a .•..ry=-,thatis,-X-=-.
Again : resuming the equations bx=a, and dy=:c, we have
hx a X ad ,, . a c a d ——- .-. -=— ; that IS, -^-=-X-. dy c y oc 0 a 0 c
These results suggest the foregoing rules.
Note.— Before performing either operation, simplify the fractions by cancelling whatever factors may be seen to be common to both num. and den. of either. If the num. of one fraction and the den. of another have a factor in common, then, before multiplying those fractions, the common factor may be suppressed.
6^ 2^ l_10a^
3^ y 2a_3a;XyXl_3a;y ^' 4^2^"6"^7~"aX5x7""35a'
(3)
a+h h a-\rh 2a_2a{a+h)
2a-\-c ' 2a 2a+c b b(2a-\-c)
/.v /o 3J2\ /a2 \ 3(a2-62) a{a-b) ^a^-b^) ^ h U{a+b)
(4) (^3a--;^(^--a^=— ^^H— ^=— ^X^^— ^— -^^.
„, 2a;— 3 Zx
,„. 6ax2 2 ex ,„
(3) ?i^x4-=.
^ ' a >/x
Examples for Exercise
-a;2 a-\-x
b^—y^ ' b-^y
(5)
x+2a ' 4a3+8a
a;2_y2 a; 1 _
%^
GREATEST COMMON MEASURE.
(7) 12.{M-2.]=. (9)
(10) (^-IH^4)=-
(11)
35. Greatest Common Measure.— A common measure of
two or more quantities is any expression that will divide them all ; and it is the greatest common measure when it is made up of all the factors which are common to the quantities measured : the symbols for the great- est common measure are G.C.M*
Let A^, B^, be two quantities in which ^ is the greatest factor common to both, so that A and B have no factor in common ; and AS being taken for the greater of the two quantities, let the successive divisions be carried on, as below, till the work stops for want of a remainder. Let the last of the quotients be d, then since Quotient x Divisor + Rem. = Dividend, we shall have the following series of equations on the right of the operation.
2)5
.-. dEz=D cD-\-Ez=C lC-\-D=D aB^C=^A
From inspecting these equations in succession, it is seen from the first that S is a factor of D, and therefore, from the second, that it is a factor of C, and therefore, from the third, that it is a factor of B, and therefore, finally, that it is a factor of A. But, by the original condition, A and B have no factor in common, except unity; therefore £"=1: hence the iinal divisor ES is simply ^, the greatest common measure of the two proposed quantities. If this final divisor should happen to be 5=1, we should then conclude that the quantities have no common measure, unity not being regarded as a measure.
The above, then, is the process for finding the G.C.M. of two quan- tities, whether they be numbers or algebraic expressions. And it is plain, from the general type of the operation, that what is the G.C.M. of the first dividend and divisor (AS, BS) is equally the G.C.M. of every other dividend and divisor to the end.
If, on arriving at any dividend and divisor, we find a factor in one which is not in the other, it may be expunged ; or a factor may be intro-
* It is very desirable that the old term measure, in reference to the present subject, should be abolished, and the term divisor always employed instead. The above symbols would then be replaced by G.G.D., which are the more to be preferred, because L.C.M. already stands for least common multiple.
GREATEST COMMON MEASURE. 25
duced into either provided it be excluded from the other; for these changes cannot affect the common measure of the two. (1) Required the G.CM. of 1x^—12x-\-5, and a^— 6a;+5 a;2-6a;+5)7a;a-12a;+5(7 7a;2-42a;+35
3003—30 or expunging the factor 30, x— l)x^^Qx-\- 5(x — 6
-5a;+5
Hence x—1 is the G.CM. (2) Required the G.CM. of ia?-2oc*-Zx+l and Zx^-2x-l. Multiply the first dividend by 3, then the second by 2, in order to avoid fractions. 3x'-2a-l)3x 4a^-6x2_9a;+3(4a>
-8x^-ix
2a;*-5a;+3)2x3a;2_ 4a:-2(3
lla?-ll, or x-l)2x^-5x-\-Z{2x-Z 2^-2-5^+3
Hence a;— 1 is the G.CM.
By finding in this way the G.CM. of the num. and den. of a fraction, and then dividing the two by that G.C.M., the fraction will be reduced to its lowest terms, so that no further simplification will be possible : thus, taking the two examples above, we have
x^-6x+5 x-5 , Za^-2x-l Sx+1
—.-z=;: ^> ^^^ ~
1x^-l2x+5 7a;-5' 4x'-2ip2_3^+l 4x24-2^-1*
the num. and den. of each of the fractions being divided by the G.CM.,
36. It may be as well to notice here that algebraic expressions are distin- guished from one another by certain designations which refer exclusively to the number of terms they contain, and to the highest power which the principal symbol in them reaches : thus ax is a monomial of the first degree, because it consists of but one term, and x occurs only in the first power ; ax"' is a monomial of the second degree, ax^ one of the third, and 80 on. Again, ax + h\s a. binomial of the first degree, ax'^-i-b a binomial of the second degree, ax'*-\-b one of the third degree, and so on. Simi- larly, Sx-—^x—] is a trinomial of the second degree, Ax^—^x' — Sx+l is a quadrinomial of the third degree, and so on. But instead of multi- plying particular names for the number of terms, the word polynomial is usually employed to denote an expression of several terms, without explicitly implying how many.
It is worth remembering that when an expression of the second degree, or a quadratic expression, as it is sometimes called, is known to be divisible by a binomial of the first degree, the quotient may be at once obtained by simply dividing the first term of the dr Jrlend by the first term of the divisor, and the last by the last : thus referring to the quotients above, we see that x—6 may be got by dividing x"- by the x, and 5 by the —-1. In like manner, Sx-j-i may be got by dividing 3^ by the x, and —1 by the
24 LEAST COMMON MULTIPLE.
—1. Similarly, knowing, as we do, that a;^— 5^— 24 is divisible by a!-\- 3 (see 33) we may get the quotient by dividing or by the aj and — 24 by the 3 ; this quotient being x—S.
If we wish to find the G.C.M. of three expressions, we find the G.C.M. of two, as above, and the G.C.M. of this and the third, and so on, if there be more quantities than three.
37. Least Common Multiple.— The least common multiple of two or more quantities is the least quantity, or the quantity of lowest degree, that is divisible by each of them. The symbols for the least common multiple are L.C.M.
If two quantities have no common measure their product is their L.C.M. ; for, since the factors of the two quantities are all different, these factors must all be contained in every expression which is divisible by them all, the least expression thus divisible is therefore simply the pro- duct of all the factors. Let, then, the two quantities have I for their G.C.M. : we may denote them by A^ and B^, where A, B, have no factor in common. It is plain that the L.C.M. will be AB^, that is, there cannot be any superfluous factor in AB^ ; for if this be divided by -45, tlie quotient is B, and if it be divided by B^, the quotient is A, and these quotients have no factor in common ; so that no factor can be spared from ABL Hence, to find the L.C.M. of two quantities, we divide their pro- duct by their G.C.M. And the L.C.M. of three quantities may be found by taking the L.C.M. of two of them, and then the L.C.M. of this and the third, and so on.
(1) What is L.CM. of 12 and 18 ? Here the G.C.M. is evidently 6.
.-.-4— =12X3=36, ihQ L.C.M. o
(2) Find the L. C. M. of x^—y^ and x''—y'^. Each of these is divisible l>y ^— 2/ (^^t- 33) • the quotient from the second isx+y (page 15), by which quantity the first is not divisible (33). Hence the G. C. M. is x — y .: {ay'—f){x'—y')-^{x-y)={x'—y\x-\-y)=x'^^x'y—xf-^y\ the L.C.M.
In a similar way, we find that the
L.C.M. of x'+l, and {x+iy is a^-\-3?+x-\-l.
„ 2x'-\-x'-2x-\, and 2x^-x''-2x-\-\ is 4a;*-5r^+ 1.
,, ic'— 1, a;— 2, and a;'— 4 is x^—5x'^-\-4t.
38. Before leaving the subject of fractions, we would invite the special attention of the student to the following property of two equal fractions : he will find it of frequent application in the solution of equations.
Let there be two equal fractions, as t=;^. Adding and subtracting
unity, we have —
3+1=^+1, that .»,—=—...[l].
a , c _ a—h c—d
^-l=-_l, „ __=_... [2].
Dividing [I] by [2], we have the property alluded to; namely, that
T- a c a+6 c-\-d , a—h c—d
If 7=^, then also — r= — -, and .'. r= .
b d a—b c—d a+b c+d
SIMPLE EQUATIONS IN GENERAL. 25
And it is plain that if, instead of the minus sign before the 1 in [2],
we had used the sign ~, implying difference, the inference would have
been, that —
„ a c ., - a+6 c-\-d , a~6 Cf-^d
If T=-5> then also, — -= -, and - = -.
0 a aryJ) C'-^d a-\-o c-\-d
39. Simple Equations in General-— In a former page of this work, some illustrations were given of the application of the first four rules of Algebra to the solution of simple equations. These illustrations were proposed at that early stage of the subject, in order that the learner might gain an insight, as soon as possible, into the use and efficiency of the symbols of algebra in certain inquiries respecting numbers. We are now prepared to take a more enlarged view of the doctrine of simple equations, and to employ methods of solution preferable to some of those adopted in the former article (art. 22).
To SOLVE A Simple Equation with one unknown Quantity. — Eule. 1 . Clear the equation of fractions, if any enter. This is done thus : re- gard every integral term as a fraction with I for denominator, and then multiply each numerator by all the denominators except its own, exactly as in finding the new numerators in the operation for reducing a set of fractions to a common denominator (34). But here the com. den. is suppressed : — the results furnishing an equation without fractions. Or, the equation may be cleared by multiplying every term by a common multiple (the smaller the better) of all the denominators.
2. Clear the equation of radical signs, if any enter. This is done thus : by transposition (20) cause the radical that is to be removed to stand alone on one side of the equation, the other terms all occupying the other side : then perform on each side the operation which is the reverse of that indi- cated by the radical ; thus, if the radical be V, square both sides; if it be '\/, cube both sides, and so on. [It is plain that the square of i^k is k, that the cube of i/k, is k, and so on, whatever k may represent : — the reverse operation merely removing or clearing the radical.]
Note. — The order in which the general precepts for the solution of an equation may be most conveniently applied will be suggested by the example itself; but as fractions are more troublesome to deal with than integral quantities, it is usual to clear them away early, as in the following specimens.
But the learner is not to expect that the neatest and best method of solving every equation that may be proposed can be explained by written instructions. He can become practically familiar with the various artifices resorted to, in particular cases, only by observing the purposes effected by them in the examples worked out for his guidance and imitation. Thus, in ex. 7, page 26, the second step in the process of solution directs that 1 be added to each side of the equation : this step is suggested from observing that, if 1 be added, the denom. of the fraction is such that, upon reducing the mixed quantity to an improper fraction, the whole of the original numerator becomes cancelled, and that thus a considerable simplification is effected.
Clearing fractions (regarding —x, and
(1) |-|+6=a;-5. (See p. 9.)
Trans. ~—^-x=: — 10,
it o
—10, as — - and —r-)
3a;— 2^— 6ic=— 60, or — 5a:=-60. Dividing by —5, a;=12.
SIMPLE EQUATIONS IN GENERAL*
(2)
■V^.-
^=0. (See p. 9.)
2 ■ 6 4 Mult, by 12, the L. C. M. of 2, 6, 4,
6;c— 6+2a:— 3a;=0. Trans., 5a;=6 . •. x=l^.
7a;+8 9:p-12 3x+1_29-8z 5
(3)
4 8 5 10
Multiply by 40, the L. C. M. of the den. 70ar+80-45a;+60=24a;+8-116+32^. Trans., 70a;-45a;-24a;-32a;=8-116 -80-60
.-. -31a:=-248.-. ;c=8.
Trans., and mult, by 2, 3v/a:=6 .'. v''^=2 ; Squaring, a;=4.
(5) v'(3+4a:)+8=2a:+9. Trans., ^(34-'4a:)=2a;+l. Squaring, Z+ix=ix'^+ix+l. Trans., 2=4:^2 . •. 4=a;' .-. a:=^i.
(6) v^(4+x)+v^a;=4.
Trans, the ^/a:, in order that v'(4+a;) may stand alone on one side, -/(4+^)=4-^ar. Squaring, 4+a:=16— 8v^a:+a:. Trans., 8v^a;=12.-. 2v/a:=:3. Squaring, 4a;=:9 . '. x=:2\.
a— a;
Adding 1 to each side,
2ax-x' "«
2 +l=t' +1, or
{a-xy -^=v'(i'+l)
{a-xr a—x
1
V{h''-\-l) H-3_7 a;-3""5' By the principle at (38), 2x 12
a ^/(^^=^+l)
(8) ^;=
(9)
^_,_e.-..=i8.
1+a;- ^ {2x-\-x') ^^^ "~1-
By the principle at (38),
v'(2a:+x') 3 . 2x-\-x' 9
-^^P^=-.-. Squaring, ^-^^-^=-.
Subtract each side from 1, then 1 ^16 , 1 _4 {\-\-xY 25' 'l+ar—S'
Reversing the terms of each fraction,
(10)
,\l-\rx=-- .' . X-. 4
5a;=-9 a:^/5-3
_1
"4*
=1.
Xy/5-^Z 2
Here the den. of the first fraction is the sum of two quantities, and the num. the difierence of their squares ; hence (p. 15) the equa. is the same as
x^ 5-3-l(a:^/5-3)=l, that is, \{x^^-Z)z=l
. • . arV'^- 3=2 . • . ar v'5=5
.•.a;=— =^5.
Multiplying by 4,
4a;+3 43— 4ar
=1
4a; that is, -^+1 3
3 129-12a:
11
=1.
(by mult, the terms of the second frac- tion by 3),
4x 129-12ar ^
••• ¥ !!-='■
Clearing fractions, 44a:— 387+36ar=0.
Trans., 80a;=387.-. cc=4—. 80
(12) a-\-x=^ {a^+x^{b^-\-x')}. Squaring, a2+2aa;+a:2=a2+a:v/ (h^'+a^) Subtracting a", and dividing by ar,
2a-\-x=s/{b*+x'^. Squaring, 4a^-\-4:ax-\-x^=P+x^. Subtracting ar^, and transposing,
Aax'=h^—ia^ . •. x=. =- a.
4a 4a
^a—<s/{a—z) *• 1-*
By art. (38), Squaring,
V{a-x)~b^'
Reversing the terms of the fractions, a-x /b-l\^ /6-l\'
—=(j+i) •*•"-"=< j+i)
a(6-l)2_a{(5+l)2_(6_i)2| • ^- (6+1)2- (6+1)2
_ 4a5 -(6+1)2
TO SOLVE A PAIR OF SIMPLE EQUATIONS, ETC.
27
(U) y{a-^^x)-\-y{a-^x)=h.
For brevity, put A for the first ex- pression, and B for the second ; then the equa. ]a A-{-B=^}); also, A^-\-B^z=:2a.
Cubing (see page 15),
A^-irB^^Z{A+B)AB=y' that is, by substitution of 2a for il^-f JB', and 6 for A-\-Bj
2a+ZMB=.l^ .
'53- 2a ^'3
Cubing, A^B^=Q^)' that is, replacing the expressions for A , JS,
—2a
(a+ ^x) (a- v^ar)=a2-a?=(^' ^^
y
.•.a;=a'— f -
(15) v'(^+9)=-
v^(^+9)-3 Add 3 to each side, then
-2.
v/(a;+9)+3=-
:+l.
^(^+9)-3 Clearing the fractions (see p. 15), a;+9-9=a;-9+ V (a;+9)-3. Trans., 12=-^(a;+9). .'.Squaring, a:+9=144.-. a;=135.
Note. — The learner is recommended to study the above solutions carefully, before proceeding to the following examples.
(1) ^_M+i=o.
Examples for Exercise. '
(12) ^(l+l)_^(l_l)=l.
(2) :r-2-(:r+l)=
Zx x-i
a: 2 1 ar_43 ^^^ 7~15~li"^6~30'
a:— 5 . x
ar-10
(5) i^%7=10.
/q\ 8x+5 7a;-3_16ar-f 15 2| ^^ 14 "^6a:+2~ 28 "^7* (10) ._!fzi(lzi)±l=i9+.^^^2
(13)
/y/a+/v''(a— a;)___l is/a—s/{a—x) a
(15) f!±^=6^-l. a^
(16) v:(!££=f5+i=5.
a— d? (17) a+^=V{a2+V(^'+*')}. l+£+^_62 1-M 1— ir+a?2~63'l— d?'
(18)
11
(11) v'(^+l)+V(^-l)=
2
n/(-^+1)
(20) ^/(a+a?) + v'(a-a?)-2V<r=0. (22) (l_.)3=l-j^.
(23) va+-^)+v(i-^)=V2.
40. To solve a Pair of Simple Equations with two unknown Quan- tities. First Method.— Rule. 1. From the first equation find an expression for one of the unknown quantities in terms of the other and of the known quantities ; that is, solve the equation for that unknown.
2. Find, in like manner, an expression for the same unknown quantity from the second equation.
a. Put these two expressions for the same thing equal to each other; and an equation, with only one unknown quantity, will be exhibited. lUis equation being solved, the value of that unknown becomes determined, and this substituted in either of the preceding expressions for the oth^
28 TO SOLVE A PAIR OF SIMPLE EQUATIONS, ETC.
unknown, completes the solution. For example, let the pair of equa- tions be —
^ ^ 3 26-5y 6+2^
7.-2y=6.-..=-^''-"^-"~T-
Clearing fractions, 182— 35y=18+6y .'. 182— 18=6y+353^ .-. 41y=164 .-. y=4
_26- 5y_6_ •'•*" 3 ~3
Second Method. — Rule. 1. Find an expression for one of the un- known quantities from either of the equations.
2. Substitute this expression for that unknown in the other equation, and the result will be an equation containing only the other unknown ; this being solved, the value of the latter becomes known, and thence, by substitution in the equation first deduced, the value of the former un- known becomes determined ; as in the following example —
3a:+6y=26^ . *. x=. — -— ^. Substituting this for x, the second
7a?— 2y=6 ) equation is ^— 2y=6. Multiplying by 3, and removing brackets,'
o
182-35y-63/=18.-. 182-18=353/+6y.-.41y=164.-.y=4.-.a?=?5^=-=2.
3 3
Third Method. — Rule. 1. Multiply the given equations by such factors — the smaller the better — as will cause the coefficients of the same unknown in the resulting equations to be equal in magnitude : whether they differ in sign or not is of no consequence.
1. If these equal coefficients have like signs, subtract one equation from the other; if they have unlike signs, add the two equations to- gether; the terms with equal coefficients will thus disappear (or be elimi- nated), and the result will be an equation with only one unknown quantity. Thus, taking the pair of equations already considered, let us eliminate y, by multiplying the first equation by 2, the second by 5, and then adding.
6a?4-10y=52 35d?— 10?/=30
41a; =82 .*. a'=:2. This value, substituted in one of the given equations, the first for instance, gives 6+5^^=26 . '. 5y=20 . '. y=4. Or, having eliminated y, we may elimi-
nate a;, in a similar manner, thus : Multi- plying each equa. by the coefficient of a; in the other, we have
21«+35y=182
21*- 6w= 18
Subtracting, Aly=16i
. '. y= 4
It is plain that the coefficients of the unknown to be eliminated may always be equalized by multiplying each by the other, as in this example : but sometimes smaller multipliers will serve the same purpose : the smallest are those which give for result the least common multiple of the coefficients.
It is worthy of notice that this third method never introduces frac- tions. As applied to a pair of equations expressed in general terms the process is as follows : —
The general equations are a^a;-\-b^y=c^, and a.je-\-b.^y=c.^, the small figures being merely marks to distinguish different coefficients. Equal- izing first the coefficients of x, and then the coefficients of y, the equa- tions become
a'O SOLVE A PAIR OF SIMPLE EQUATIONS, ETC.
59
1st.
(a, (a,
{a yh^—aj!) ^y=a^c^—a^^
y=
2iid.
boCi — J,c
These general expressions for the unknown quantities comprehend, of course, every particular case. In any specified example, if the given co- efiBcients be substituted for the general coefficients above, the solutions for that example will be obtained. And if either of the other methods be applied to the preceding equations, the same general values will be deduced.
It may be noticed here, that every such general expression for an un- known quantity is called a. formula. The above are the tvio formulcB for the solution of a pair of simultaneous simple equations with two unknowns. Equations are said to be simultaneous when they coexist for the same values, in each, of the unknown quantities. Such equations, to admit of definite values for these unknown quantities, must be independent of one another ; that is, each must imply conditions distinct from those implied in the other equations. If one of the equations be but a necessary con- sequence of some other of them, that one cannot be regarded as embody- ing a distinct condition, so that the number of independent equations will be fewer than the number of unknowns to be determined from them ; and to fix the values of these unknowns another condition would be necessary.
We shall now give a few examples of the solution of a pair of simulta- neous simple equations.
(1) 2a;-f 5y=23 3a?— 2y= 6 Multiplying the first by 3 and the second
by 2, to equalize the coefficients of «, 6iB+15y=69 6x— 4y=12
By subtracting, 193/=57 . *. y=3.
This value of y, put for that symbol in one of the given equations, determines x ; thus, from the second equation,
dx-6=6 .: 3x=U .'. x=L (2) Zx-\-^yz=Z6\ f9x-\-y=10S i(3y-a.)= 4f °^' iSy-x= 16 From the first, y=10S—9x Substituting this in the second, 324-27*-a!=16 .-. 28a;=308 .-. a=ll, .-. y=108-9^=9.
ix+y=5%( ^^' Ux-\-ky=17 By subtracting, (|— J)a;=5. Mult, by 12, (9-4)a?=60.-. a?=12 .'.ix+iy=i-\-y=12.:iy=8.'.y=16.
(4) i+-=12^ (^+y=12
'-'-= ij U.'-4y=i
X y
where x\ t/, are put for -, -. X y
Multiplying the first by 3, and sub- tracting, 7y =35 . '. y=5 . •.a;'=12-y=7 11 11
(5)
.-..=-=-, and ,=^-.
5 ^4
6x-2y y_^^
3 6
Clearing fractions, these become 17a;+16y=200\ f l7a;+16y=200 12x-By=84: j °'*' \ 4a;-y=28 From the second, y=4a3— 28 ; and by substi. in the first, 17x+ 64a;- 448=200 . •. 81x=648 . •• x=8 .-. 2/=4x— 28=4. Otherwise, Multiplying the second equa. by 16, and adding to that above it,
30
TO SOLVE THREE SIMPLE EQUATIONS, ETC.
81a;=200+448=648 .-. a;=8 .-. 2^=403-28=32-28=4.
(6) a{x-y)=h{x-\-y) [1]
x^-y^=c [2]
Midt. [1] by £c+y, and [2] by a,
<i{x^—y^)=zac
0 = l{x-\-yY—ac
Consequently, from [1],
f3a;+42/=43 ^^' \lx-2yz=21.
nx-zy=-n
^^' t7a;+2y=81.
f3x+4y=29 ^^^ \17x-3y=36.
6 /ac
Adding and subtracting,
J\ /ac a—b /ac
~b But ^"'- /«^^_V«^^'^
V -62-— J
Examples
•^=2^\/«^^
FOE EXEECISE.
{Zx-7y 2x-\-y-\-l
(4)
(5)
n(a;+2) + 83/=31 U{y+5)+10x=192.
'a;+y
+1=6
^^3=4.
2a;- 3
(6) ^ 2 ■+^=^ 5x-lZy=.ZZl.
(7)
f3^-% 2:^+y
"~2~+^— 6~
a;-2y a; «
^ — r-=2+3-
(8) .
(9)
(10)
(12)
3 I
l+i=5
a; y
1-1=1.
\x y /'4 6_17 a: y 63
a; 3/ a;+y=20
120. J2a;+-42/=l-2
<") {?4=
[3-4a;— •02?/='01.
(13) V(4^+1)-n/3' ( 9a;— 5y=5. /V£+6_N/y+2
(14) V^+4 Vy-2 i 2 V^+ 3^3^=2.
41. To solve three Simple Equations with three Un- known Quantities. — In solving a pair of simultaneous simple equations, it has been sufficiently seen that the first object is, by the elimination of one of the unknowns, to reduce the two equations to one, containing only the other unknown. In like manner, in treating three simultaneous equations, the aim is to reduce them to two, by the previous elimination of one of the unknowns ; this is done as follows.
Rule. 1. Take any two of the proposed equations, eliminate from them one of the unknowns: the result will be an equation containing only the other two.
2. Take now the third equation, in connection with one of those already employed, and eliminate tlie same unknown from them : the result will be a second equation containing only the other two unknowns. We shall thus have reduced our three equations with three unknowns, to two
TO SOLVE THREE SIMPLE EQUATIONS, ETC.
31
with two unknowns; and these may be solved as already explained; and thence the value of the third unknown found by substitution, as in the following examples : —
|
f2x-\-4:y—dz: |
=22 |
. . . [1] |
|
(1) ■ ix-2y+5z |
=18 |
. . . [2] |
|
M-^7y-z= |
=63 |
. . . [3] |
|
Kliminate x from [1' |
and |
[2], thus |
|
4x-\-8y-6z= |
i44 |
|
|
Ax-2y-\-5z= |
=18 :26 . |
|
|
lOy-ll0= |
. . [A]. |
|
|
Eliminate the same |
from |
[1] and [3], |
|
thus 6x+12y-9z= |
=66 |
|
|
6x+7y-z = |
=63 |
57/-8z=. 3 . . . [B].
We have thus two equations [A] and [B] with only two of the unknowns.
Multiplying [B] by 2, and subtracting, to eliminate y, we have 52=20
3+82
.'.z=i,y=-—-=7, x=
22- iy-\-Zz
5 - 2 ='
where y is got by substitution in [B], and X by substitution in [1].
(2)
{2x-^2y—4z=S \ 6a;— 3y+3z=33 [7x+ y -[■5z=65
. . . [1] . . . [2] . . . [3]
may be most
Here it is plain that readily eliminated.
Subtract [1] from twice [3], then lla;+142=122 ... [A]
Add [2] to three times [3], then 26a;+182=228 or, 13a;+92=114 . . . [B]
Mult. [A] by 13, and [B] by 11, 13.11^+1822=1586 11.13a;+ 992=1254
832= 332
122-142 ^ .'. 2=4, x= — ^^; =6, and [3],
/=65-
11
.7x-
•52=3.
Examples for Exercise 'x-\-y-\-z=i5B
(1) ■ a;+2y+ 32=105 la;+3y+42=134. 'x-\-y+^z=z27
(2) ■ x-\-y+iz=20
x-{-ly-^\z=:16. / 3a;- 9y+ 82=41
(3) ] 6:c- 43^-22=20 lll;z;-7y-62=37. 'x-hly-\-hz=Z2
li^+
(4) i h^+iy-^^z--
(5)
(6)
0)
f2x—2y-{-5z=27 \ Sx+6y-4z= 2 l5a;+4y+22=40.
/ 5a;- 4?/+ 22=48 J 3a;+33/-42=24 (2x-5y+Sz=19.
(8) i
^+-=1 X y
X z
:15
-^y+^2=12. f2a;+3y+42=16 ] 3a;+2y-52= 8 (6x—6y-\-Zz= 6.
It was observed (at p. 29), that in order that a set of simultaneous equa- tions may admit of definite solutions, the equations must be independent of one another : they must, of course, be also compatible with one another ; that is, no two of them may involve a contradiction. In the first of the following sets, the third equation is an obvious consequence of the first, so that the conditions are insufficient : in the second set, the conditions are contradictory, for if twice the first be added to the second, we shall have 8a;— 32/ + 2«=38, whereas the third condition affirms that this is 35, so that 38=35, which is absurd.
Insufficient Conditions. 2.r-33/+42=12 6x-\-2y-z=14 4a;-6y+8«=24
Incompatible Conditions. Sx—2y+5z=U 2x-\-y-8z=10 Bx-3y+2z=35
32
TO SOLVE THREE SIMPLE EQUATIONS, ETC.
But we need never take the trouble actually to examine a set of simul- taneous equations with a view to discovering whether they are indepen- dent and consistent or not : we may proceed at once with our elimi- nations. Indications, abundantly obvious, of insufficient or incongruous conditions, will spontaneously offer themselves. Thus, taking the first set above, if we eliminate z between the first and second, we shall have 26a; + 5i/^68, and if we then eliminate z between the second and third, we shall have aho 26;»+52/=68; and we at once infer insufficiency of conditions. As to the other set of equations, we have already seen that they involve the absurdity of confounding 35 with 38.
By the process of successive elimination just explained, any set of simultaneous equations, however numerous, may be solved.
Questions to be solved by Simple Equations.
Note. — In solving questions by algebra, instead of invariably repre- senting an unknown quantity by x or y, it will often be better to denote it by some multiple of the letter ; that is, to prefix to it a coefficient. Whenever a fractional part of the unknown is to be taken, it will obviously save work and trouble if we prefix to the letter a coefficient, such that the prescribed fractional part will be a whole number (see exam- ples 2 and 3 following).
(1) A labourer was engaged for 60 days, on condition that for every day he worked he should receive \^d., and for every day he idled he should forfeit 5^. At the end of the 60 days he received 20s. : how many days did he work ?
Is^ Solution. Suppose he worked x days, and idled ^ days, then by the question a;+2/=60 and 15a;— 53/=240, pence 5 times 1st equa., 6a;-|-5y=300
.•.20a; =540.-.a;=2r
Hence he worked 27 days.
2nd Solution. Suppose he worked x days, then he idled 60— a; days, and by the question
16a;— 5(60— a;)=240, pence, that is, 15a;-300-i-5a;=240
Trans., 20a;=540 .*. a;=27
Hence he worked 27 days, and conse- quently he idled 33 days.
Note. — In the solution of pro- blems, it is usually preferable to employ as many unknown quan- tities as there are values to be found.
(2) A gamester at play lost J of his money, and then won 10s.: he
afterwards lost J of what he then had, and won 3s., leaving off with 3 guineas. How much had he at first?
Suppose he had 5x shillings at first : then he lost x shillings ; so that after winning lOs. he had ix-\-10 ; of this sum, he had at last only two -thirds and 3s., which by the question was 63s. .-. |(4a:-|-10)+3=63.-. |(4a:-f 10)=60 .-. 8a;-|-20=180 .-. 8a;=160 .-. a;=20 "5a:=100. Hence he began with 100s., or £5.
(3) Two persons, A and B, have the same annual income : A saves the fifth part of his ; but B, who spends £80 a year more than A, at the end of 4 years finds himself £220 in debt : what is the annual income and expenditure of each ?
Let 5a; represent the income, then A spends 4a;, and B spends 4a;-|-80 yearly, .'. in 4 years B spends 16a;-|-320, while his income in that time is only 20a; : hence by the question, 16a;-f 320=20a;+220. .-. Trans., 100=4a;.-.a;=25 .*. 6a;=125. The annual income is .*. £125. Also, 4a;=£100, J.'s expenditure, .♦. £100-f £80=£180, -B's expenditure.
TO SOLVE THREE SIMPLE EQUATIONS, ETC.
33
(4) There is a certain number consisting of two figures, the sum of which is 5 ; and if 9 be added to the number the figures will be reversed : what is the number ?
Let X , y represent the figures, then the number is V)x-\-y\ but if the figures be reversed the number is \^y-\-X'. hence by the question,
a:+y=5 .*. ?/=5 —x
Substituting the above value of y, 10a?+5-^-|-9=50— 10<i?+a?. Trans. 18d?=36 .'. x= 2 .'. y=5— d?=3. Hence the number is 23.
(5) There are two sorts of spirits, one worth 20s. a gallon and the other worth 1 2s. : how much of each must be taken to make 1 gal- lon worth 14s. ?
Suppose X gal. at 205. and y gal. at 12«. : then by the question,
X-\-y=\.'.y=.\—x. Also, the worth of the compound is
20a?-|-12y=14, .*. by substitution,
20^-fl2-12*=14. Trans., ^x=.1:.x—\:.y=.\—\-=\, so that there must be \ gal. at 20s., and I gal. at 125.
(6) Spirits at 9s. 6^. per gallon were mixed with some at 13s. 6c?. There were 104 gals, in the mix- ture, and they were worth £56 : how much of each sort was used ?
Suppose X gals, at 9s. 6f?., and y at 13s. 6(i. : then, in sixpences the worth of the mixture is
19a?+27y=2240 (by quest.) Also, x-^ y z= 104.-.a?=104— 3/ .-. 19^+19y=1976
8y=264.-.y=33.-.^=71. (7) A labourer is hired for a spe- cified number of days, on condition that for each day he works he shall receive a certain sum, but for each day he is idle he shall pay a certain sum for his board. At the end of the time the proper balance is paid to him : how is the number of days he worked to be found ?
Let a=the price of each day's work, 5= ,, ,, board,
c=the amount received, c^=:the whole number of days,
Also d?=the number of days' work.
Then the price of the work done is aa?, and that of the board h {d—x).
By the question, the difference of these is ax—h{d—x)=.c,
or ax-\-bx—ld=.c .'. {a-\-h)x=zc-\-'bd
c-\-hd . , ad—c .'. X = -, and .*. d—x= -,
the former value expressing the number of working days, and the latter the number of idle days.
The above is a general solution of the problem of which ex. 1 is a particular case. Putting in the preceding formulae a=15, 6=5, c=240, and d=60, we have
240+5.60 540 „^ , , ^. „„
The second formula above enables us to find the number of idle days, without first finding the number of working days,
(8) If A and B can finish a piece of work in 8 days, A and (7 in 9 days, and B and C in 10 days : in how many days can each, alone, finish it?
Let X, y, z represent the number of days occupied by ^, B, C, alone; then in one day, the parts of the whole they can
do respectively will be -, -, and -, which, X y z
for brevity, write a/, 3/, and 2'.
Now, since A and B together do the
whole in 8 days, A and (7, in 9 days, and
B and C in 10 days,
.'. 8*'4-8y=l (one whole work)
9a?'+93'=l/.a?'+2'=i
ioy+io/=i.
Mult. 1st by 5, 3rd by 4, and subtract,
.-. 40y-402'=l .-. a?'-s'=— . 40
Add and subtract the two equations on
the right,
. 49 . 31
720'
720
84
TO SOLVE THREE SIMPLE EQUATIONS, ETC.
Substitute these in the third equation, . ,. , . 31 41 ,41
. ^20 ^ /4 ^ . ^^ ,
. . «'=-j^=14— days required by A.
720 ,„23
_720_ 7 2— 337-23gj „ „ a
(9) It is recorded that Hiero, King of Syracuse, ordered a crown of solid gold to be fabricated for Jupiter: its weight was to be 10 lbs. (avoir.), which weight the crown was found exactly to balance. But, suspecting that there was an alloy of silver in it, he directed Archi- medes to ascertain the extent of the fraud. The philosopher found that a specimen of the gold supplied,
52 when weighed in water, lost jr— :
of its weight ; and that silver lost
-— - of Its weight m water; also.
that the crown itself lost 10 oz. of its weight. How much of each metal was there in its composition?
Let X be the ounces of gold, and y the ounces of silver, then a?+y=160 ; also,
— -— = what the x oz. of gold loses in water,
993/ and — — = what the y oz. of silver loses
in water ; and the sum of these is the
whole loss of weight of the crown in water :
52a; 99y •■•lOOO+lM=l<'---52x+99^=10000.
From the first equation, a;=160— ?/.
Hence by substitution,
8320-52y+99?/=10000.
,- -^on 1680 „ 35
.-. 473/=1680 .-. 2/=— —=35— oz. 47 47
.12
.-. ic=160-y=124— oz.
.12
Hence the crown was composed of 124 — oz.
47
of gold, and 35 t^tOz. of silver.
Questions for Exercise in Simple Equations with One, Two, and Three Unknown Quantities.
(1) Divide £100 among A, B, C, so that A may have £20 more than S, and B £10 more than C.
(2) A farmer has wheat at 4s. M. a bushel, with which he wishes to mix rye at 3s. dd. a bushel, and barley at 3s. a bushel, so as to make 136 bushels that shall be worth 4s. a bushel : what quantity of rye and barley" must he take ?
(3) What fraction is that to the numerator of which, if 1 be^dded, the value will be ^ ; but, if 1 be added to the denominator the value will bei?
(4) A man and his wife usually drank a barrel of beer in 15 days, but, after drinking together from a fresh barrel for 6 days, the man left home, when the woman alone drank the remainder in 30 days. In what time could either alone finish the barrel ?
(5) A person is desirous of relieving some poor persons by giving them half-a-crown each, but finds he has not money enough by 3s. : he there- fore gave them 2s. each, and had 4s. to spare. How much had he at first, and how many did he relieve ?
(6) Divide 90 into four parts, such that if the first part be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the four results may all be equal.
(7) The trinomial expression ax'^-^ba;-\-G is such that when 4 is put for a?, its value is 42 ; when 3 is put for x, its value is 22 ; and when 2 is put for x, its value is 8. Find the values of the coefficients a, b, c.
TO SOLVE THREE SIMPLE EQUATIONS, ETC. 35
(8) A hare is 80 of her leaps in advance of a pursuing greyhound, and she takes three leaps to the greyhound's two ; but one leap of the latter is equal to two leaps of the former. How many leaps will the hare have taken when she is caught?
(9) In what time would the work alluded to in ex. 8, p. 33, be done by A, B, and C all working together ?
(10) Gunpowder is to be conveyed into a garrison in full casks : more than 2 cwt. cannot be carried in each. There are two sorts of powder : a cask full of one sort weighs 230 lbs., a cask full of the other weighs 182 lbs. How much of each sort must be sent to make a cask full of the extreme weight ?
(11) A cistern can be filled by the pipes A and B in 70 minutes, by the pipes A and C in 84 minutes, and by the pipes B and (7 in 140 minutes. What time will each pipe take to fill it, and in what time will it be filled if all the pipes flow together ?
(12) A composition of copper and tin, containing 100 cubic inches, weighed 505 oz. How much of each metal did it contain, supposing a cubic inch of copper to weigh 5^ oz. and a cubic inch of tin to weigh 4i oz. ?
(13) 37 lbs. of tin is found to lose 5 lbs. when weighed in water ; and 23 lbs. of lead to lose 2 lbs. when weighed in water ; a composition of tin and lead, weighing 120 lbs., is found to lose 14 lbs. when weighed in water. How much of each metal is there in the composition ?
(14) A number consists of three figures, such that the difference between the first and second is the same as the difference between the second and third. If the number be divided by the sums of the figures the quotient will be 26, and if 198 be added to the number, the figures of it will be reversed. Required the number.
(15) Three ingots, A, B, 0, are composed of different metals: 1 lb. of A contains 7 oz. of silver, 3 oz. of copper, and 6 oz. of pewter ; 1 lb. of B contains 12 oz. of silver, 3 oz. of copper, and 1 oz. of pewter; and 1 lb. of C contains 4 oz. of silver, 7 oz. of copper, and 5 oz. of pewter. How much must be taken from each ingot to form 1 lb., which shall con- tain 8 oz. of silver, 3f oz. of copper, and 4^ oz. of pewter?
(16) Standard gold for coinage is £3 17s. lO^c^. per oz. What is the smallest integral number of ounces that can be coined into sovereigns, and how many sovereigns will there be ?
(17) The hands of a clock are together at 12 o'clock. State all the times at which they are together from 12 o'clock at noon till 12 o'clock at night.
(18) Three vessels together contain 18 gallons: half the first is poured into the second, a third of what is then in the second is poured into the third, and, lastly, a fourth of what the third then contains is poured into the first; the liquid is then found to be equally divided among the three vessels. What did each contain at first ?
(19) A, B, C have certain sums of money, such that if A's were increased by half what B and C together have, he would have £a ; if i^'s were increased by one-third of what A and C have, he would have £b ; and if C's were increased by one-fourth of what A and B have, he would possess £c. What sum has each ?
(20) If from a vessel of wine containing a gallons, b gallons be drawn off, and the vessel then filled up with water ; then b gallons of the mix- ture be drawn off and the vessel again filled up with water, and so on ;
D 2
36 FBACTIONAL EXPONENTS.
and that this operation be repeated n times successively : what will be the quantity of wine left after these n drawings ?
(21) Two gamesters play on the following conditions: whichever loses the first game shall double the money of his opponent ; whichever loses the second game shall triple the money of his opponent ; and whichever loses the third game shall quadruple the money of his opponent. They lose alternately, and at the end of the third game each has the same sum, namely, 48s. How much had each at first?
(22) Five persons engage in play on condition that he who loses shall give to each of the others as much as that other already has. All lose in turn, and yet at the end of the fifth game they all have the same sum, namely, £32. How many pounds did each begin with ?
42. Theory of Exponents. — It has already been explained (27,
28) that when n is a positive integer a" denotes the wth power of a, and a" the nth root; we are now to extend this notation, and to give a consistent meaning to a quantity with any exponent, whether whole or fractional, positive or negative.
Negative Exponents. — The extension to the case of a negative exponent is suggested from the following considerations : —
=1 .'.a^ :=1, whatever be a
.1 . ^_i ^1 a ' ' a
1 -2 1
=-2-''« =-2 a a
Consequently, whatever integer n may be, a"" is the same as — ; 1,
divided by any quantity, is called the reciprocal of that quantity, so that, not only powers, but also the reciprocals of powers, may be indicated by exponents : in the former case the exponents are positive, in the latter, they are negative.
43. Fractional Exponents. — From what has already been laid
down, we know that (a*")" means the nth root of the with power of a (27, 28). Instead of this cumbersome notation, the following more com-
pact form is used, a" ; it being agreed that the numerator of the ex- ponent shall denote the power, and the denominator the root of that power. It is matter of indifference, however, whether we call the ex- pression just written the nth root of the mth power of a, or the mth power of the nth root of a\ for in whichever order the operations are performed upon any number or quantity, the final result is the same;
that is to say, {a"") " and \a"/'" are the same in value. For instance,
|
By division (32) — = |
-a^-l |
=aO |
|
a |
||
|
a' 3 |
=a}'^ |
=a-l |
|
a |
||
|
=a^-3 |
-a-2 |
|
|
a |
||
|
a' And generally -r— |
n |
•»=a-» |
|
a ■*" |
|
But |
1 a |
|
but |
a' a' " |
|
but |
a' |
|
but |
a' |
SQUARE ROOT OF A POLYNOMIAL. 37
suppose 77i=:3 and n=5 ; then we have (a^)^={aaaf^=(fa^a^, the 5th power of each being aaa; and since {J)^=:a^Ja\ consequently, (a^)^
44. The following principles require to be established before we can safely apply the fundamental rules of operation to quantities with ex- ponents : —
r m r n s
,y- ...
1. a!xx...to m factors Xyyy... to m f actors =a?y.a?5/.ary.... to m factors.
^ xxa; ...to m factors x oo x
2. ;: =-. — . -. ... to m factors.
y^/y... to m factors y y y
(vm _ _ _^_ _ _
*\ n '"' . s n s . n s
Consequently, a?'»X3/'"=(a?y)"*, and — =( - )
3. Put the single symbols a. and 5, for a;" and a;* ; then (a?**/ =a'*, and \x^) =■
6*; that is, a?'»=a'», and a?'' = &*.*. a;'«*=a% and ««'•=&"« [A]
.-.ar^'X «"'■=«»* X 6*", that is, a?»"*+«''=(a&)'**.-.a&=a? «* , that is, replacing the symbols, a, 6, by their values above, a)" Xa;*=a? '** =«;'' * .
4. From the equations marked [A], ic'»«-^a;'"■=a«*-7-^'''*=( j ) ; that is, a;'"*-«''=
»w— nr
(a-T-J)"* .'. a-7-6=a; "* , or replacing the symbols a, 5, by their values,
m r
Hence, whatever be the exponents attached to equal quantities, the fol- lowing are the general rules for multiplying and dividing.
45. For Multiplication. — Add the exponents.
For Division. — Subtract the exponent of the divisor from the exponent of the dividend.
As to Powers, we merely multiply the exponent by the index of the power; thus (a;'")"=^""', because each form equally represents :c'" re- peated n times.
p mp £
For Roots, the rule is the same; that is, (^'")"=^" ; for (;»'")'• =
mp
(jafpf , by last case, and by (43), this is the same as i» " .
Examples,
(1) JxJ^=a^'^^=a^. (2) a*Xa~W~^=a^=a%*=aV«-
(3) a-*^a^=.-^-U-^=l= J:,. (4) (.^)^=A
(5) (.-4)i=a-t=4=JL' (6) Q.'')-^=J=^a^.
46. Square Root of a Polynomial.— In this article we pro- pose to show how the square root of any expression of even degree of the
38 SQUARE ROOT OF A POLYNOMIAL.
form A-\-BoB-\-Cx^-^Dx^-\-Ex^-\- . . .*, may be extracted. We shall find, by observing the constitution of a polynomial, known to be the square of a given root, that the reverse operation of evolving the root from the square will be suggested.
1. The polynomial x''-\-'>lax-{-a^ is known to be the square of x-^a, and it is easy to see that this root may be evolved from its square thus ; the square-root of the leading term is x, the first
term of the sought root ; and x^ being subtracted x'^-\-2ax-\-a'\x-\-a
from the proposed trinomial, leaves 'Hax-^-d^ for • x^ remainder, the leading term of which, if divided 2a;-f a] 2ax+cfi by twice the first term x of the root, already ob- -' 2ax-\-a?
tained, will give the second term a. We may regard
^x as a trial or partial divisor of the remainder
^ax-^a^y and place it against this remainder, as
in the margin ; and having from the trial divisor got a, we have only to
connect this a with the trial divisor to get the true or complete divisor
due to the quotient a.
2. The polynomial x^ -\-':iax^ +{a^ ^^h)x^ J{.^abx4-h'^ is the square of x'^-{-ax-^h. And proceeding in imitation of the process above, doubling what may already be in the root-place, for the trial divisor expected to give the next term of the root, we find the complete root as before, thus : —
2a;2+a^| 2ax^+ (aH 26)a^ 2ax^+a'^x^
2lx^-\-2abx-\-W-
47. There was scarcely any necessity to exhibit this last operation ; the efficiency of it might have been inferred from the preceding one; for since(;»Ha^-f&)2={(rcH«^) + 6F=(^' + «^f + 2%' + «^) + fc'(p.l5),the proposed square is no other than the trinomial just written. The root of the first term, that is the portion x'^-\-ax, of the whole root is got as pre- viously explained, and thence the remaining portion h. And by the same uniform process is the root of the polynomial furnished by {x^-\-ax'-\-hx -\-cf or (x^-\-ax^-^'bxY-\-^c{x^-\-ax--\-hx)-\-c'^ obtained; and so on: the rule for the operation is, therefore, this : —
JRuLE. 1. Arrange the polynomial as if for division.
2. The square root of the first term, will be the leading term of the sought root ; place the square of it under the first term, to which it is of course equal, and, having drawn a line under it, bring down the next two terms of the polynomial ;i- regard these as a dividend, and for a trial divisor of it, write twice the root-term just found.
3. Find the quotient of the leading term of the dividend by this trial divisor, and connect it, with its proper sign, both to the root-term, and to the trial divisor ; the divisor will then be completed.
* It is scarcely necessary to premise here that x may be anything — even 1. + When terms are absent from the arranged polynomial, their places should be supplied by zeros: thus x'^-\-Qx"-\-2x-^l should be written si^-\-(ix^-\-Qa?-\-2x-]rl,
CUBE ROOT OF A POLYNOMIAL. 39
4. Multiply the complete divisor by this second root-term ; subtract the product from the dividend, and connect with the remainder two more terms of the polynomial for a second dividend.
5. Proceed now exactly as before, taking twice the part of the root already found as a trial divisor, for finding the third term of that root, which third term, when thus obtained, completes the divisor; and so on till all the terms of the polynomial have been brought down.
If after this, there be still a remainder, we may be sure that the poly- nomial is not a complete square.
(1) ix^-ix^-\-ldx^-6x+9\2x^-x+B (2) 4x*-12x^+25x^-2ix-\-lS\'2f-Zx-{-4: ix* 4a;*
4x^-x\ -4^3+13^^ 4a;2-3£| -12a;3-f 25a;2
~ _4a;3-fa;« -12^3+9^
4x^-2x-i-d\ 12x^-6x+9 4a;^-6a;-f4| 16x^-2ix+18
12x^-6x-\-9 16x^-2ix+16
We infer from there being a remainder in this second example, that the proposed polynomial is not a complete square.
Examples for Exercise.
(1) Ax*-As(^-dx^+2x-\-l. (2) 9x*-eax^-\-a^. 3) 9x^-12a^+10x*-2S3(^-\-17x^-8x-\-lQ. (4) Ax^+12x^-\-5x*-2x^-{-1x^-2x-\'l.
(5) ix'y*-12a^/-{-17x*f^l2x^y-\-ix^ (6) ^+2L^+2(-+^)+3.
48. Cube Root of a Polynomial.— Examining the constitu- tion of a cube, we find that
{x-]-af=x^-^{2x^-\-Sax-{-a^}a [See page 15].
&c. &c.
Taking the first of these cuhes namely, a?-\-Za^'^Za^x-\-a^, the process in the margin, for evolving the root, is suggested.
The first term x of the root being found, a?-\-dax^^da^x-\-a^\x+a
and the dividend, consisting of three terms, „ '-
brought down, we write 3 times the square _
of the root-term as a trial divisor for find- Zx'^-{-Zax a^ Zax^-{-Sa^x+a^ ing a ; we then complete the divisor by an- Sax^-\-Sa^x-\-a^
nexing three times the product of the last — —
found term and the preceding, and also the square of the last found term. Hence the ollowing rule.
Rule. 1 . The terms being arranged as in the square root, find the root of the first term, place the cube of it under that term and brmg down the three next terms for a dividend, the trial divisor of which will be tbree times the square of the preceding part of the root.
2. By aid of the trial divisor find the second term of the root, and complete the divisor by annexing thrice the product of the last root term and what precedes it, as also the square of this last root term.
3. Multiply the divisor by the new root term, subtract the product trom the dividend, and to the remainder annex three new terms from the poly- nomial. And proceed, as before, to find a third root term; and so on.
40 IMAGINARY OR IMPOSSIBLE QUANTITIES.
Ex. 27x'^-5ia^-{-6Bx*-Ux^-\-21x^-6x-\-l\Sx^-2x-{-l
27a;4-18a;3+4^| -5ix^-\-eSx*-Ua^ ~~ ^5^x^-\-Z6x*- 8x^ 27x*-B6x^-{-21x^-ex+l\ 27;t*-36a;3+21a;2 -6x+l =d{dx^-2xy-Z{Zx^-2x)-j-l 27x*-B6a^-\-21x^-6x+l
Examples for Exercise.
(1) x^-ex^^-^-lBx* -20x^+15x^-6x^1. (2) x^+Qx^-^i0s^-^96x-Qi.
(3) 8aP-{-d6x^-^5ix-j-27. (4) x'^-3ax^+6a^x^-7a^x^-\-Qa*x^-Za^x-^a'^.
From the preceding general rules are derived the methods for extracting the square and cube roots of numbers, as given in most books on Arith- metic. But as respects the extraction of the cube root of a number, a much more simple and convenient process will be explained hereafter.
49. Surds. — ^When a quantity having the radical sign, or a fractional exponent attached to it, is such that the indicated root cannot be accu- rately extracted, the expression is called a surd, and sometimes an ir- rational quantity. Quantities which are not surds are called rational quantities. The following are all surds, namely, a/2, -v/5, 1^/7, -v/S, &c., &c., because there is no number whose square is accurately 2, or 5, or 7, or 8, &c. ; so also \/G, \/8, &c., are surds, because there does not exist any three equal numbers which, multiplied together, will accurately make 6, nor four equal numbers which, multiplied together, will make exactly 8, &c. The above are arithmetical surds: the following are
algebraic surds; ^/a, V^^ V^^' (^+^)^' <^c-» because no letter, or letters, free from radical signs, can, by the reverse operation of squaring, cubing, &o., produce a, a^, &c.
On the contrary, ^/4, ^/9, ^8, V"" ^^' V«^ ^/«^ •^{a+h)\ &c., are all rational quantities : the indicated operations can be accurately executed, and the results correctly exhibited: they are 2, 3, 2,-3, a^, a*, [a-^bf, &c.
50. It must be noticed, however, that the result of a square root ope- ration upon a positive quantity is, in strictness, of a twofold character : it is either +, or — : thus v 4 is as much —2 as +2, because, which- ever of these we multiply by itself, the result is still 4 : hence, employing the double sign, we are justified in saying that \/4=±:2, that is, that the square root of 4 is either plus or minus 2. In like manner, -s/9=±3, v/36 = ±6, and so on. And this twofold form equally belongs to every even root of a positive quantity, because an even number of factors, whether they be all positive, or all negative, equally gives a plus product ; thus, ^16 = ±2, because (2)*, and (—2)'' equally give 16. Any odd root of a quantity is, however, unambiguous, because any odd number of factors, with the same sign, produces that sign: thus V8=2, y — S— — '2, &c., because 2-^=8, (—2)*=— 8, &c. : and it is plain that no other real values of V^, V-8, &c., exist.
51. Imaginary or Impossible Quantities.— But when an even root of a negative quantity is indicated, there is implied an impossi- bility ; such root is called an imaginary root : no even power of a number, whether the number be positive or negative, can ever produce a negative result: thus, such operations as V— 4, ^—16, is/ —a, &c., are all of
REDUCTION OF SURDS. 41
impossible performance : there is not only no number which, multiplied by itself, will produce —4, but there is no number which, squared, can even approximate to —4. It is true that no number, squared, can pro- duce 2, or 5 ; but we can assign a number which, when squared, shall approach as near to either of these as we please, as arithmetic shows : thus, v^2=l*4142...., and \/5=2'23606... But imaginaries cannot even be approximated to. All that can at present be said, as to the interpre- tation of imaginary quantities, is, that whenever they necessarily enter into the result of any problem, that problem may be safely inferred to be of impossible solution, or to involve impossible conditions.
5-2. It is desirable to notice that every single imaginary may be always represented by two factors, one of which is real, though not always rational, and the other the imaginary, -s/ — 1 oyX/ — 1, &c. :* thus ^ —A
= V(4X-1)=2>/ — 1, V-16=2V— 1, ^-5 = ^5^^ — 1, V'-«= V'a^/ — 1, &c.
Especied care, too, must be taken not to regard such expressions as /v/( — 5)-, V(— 2)*, ^(^af, &c., as imaginary: in these, the operations indicated by the radical and the exponent, merely neutralize one another, and the quantity to which they are attached remains unaffected by their joint influence: thus, ^(-5f=-5, V(-^)'=-2, v/(~a)^=— a, &c.
2. 4 2.
In fact, these expressions are the same as (—5)2, (—2)^, (— «)^, &c. (43).
53. Reduction of Surds. — A surd may be reduced to another of simpler form, whenever the quantity under the radical has a factor upon which the extraction of the root indicated can be actually performed : thus, since ^3 8= ^(9 x2).•.^/18=3^/2. In like manner, since ^(a^— a^^)= ^{a\a^a;)] .-. j[(i^—a'^x)=a^\a—x). If the surd be fractional, it can be reduced to an integral form by multiplying numerator and denominator by such a quantity as will make the new denominator a complete power corresponding to the root : thus, —
T 1-1 / ^^ / 4.2a;^a; 1x /2x 2x /Qxy_ 2x
In hke maimer, ^^^^=^__^=— y-=-^y ____.^6.:2,.
Examples for Exercise.
r2a^
a) Vh
(2) ^60=.
(3) ^-54=.
(4) ^/12aV/=.
(7) {Sa%^-^=.
(8) ■i/{ax'+!>x')=.
(9) ^{Za'x+eabx+Zt'x)—
(10) («+-)v/|^=-
54. Besides reducing a surd to its simplest form, it is necessary, when two surds are to be multiplied or divided,' the one by the other, that both be reduced to a common surd-index. This is done by using fractional exponents instead of the radical sign, and then bringing the fractions to a common denominator, which denominator is, of course, the surd-index common to both expressions : thus, the surd-indices in ^a and ^^b, are
♦ It may be proved, though not conveniently in this place, that any root of —1 is reducible to a form involving no imaginaiy, but y/ —1.
4<2
BINOMIAL SURDS.
different, namely, 2 and 3 : to change the surds into equivalent ones with the same index, we change the fractions i, J, in a^, b\ into their equiva- lents f, I : the expressions are then a«, b^, or \/a^, \/fe^ so that ^a, X/b, are thus reduced to equivalent surds with a common surd-index, namely, the index 6.
55. Addition and Subtraction of Surds.— EuLE. l. Reduce the surds to their simplest forms. 2. If the surds be like or similar, add or subtract the coefficients, and annex the common surd part to the result.
But if the surds are not similar surds, the addition or subtraction can only be indicated —
(1) ^/72+v^l28=,/(36x2)+^/(64x2)=6^/2^-8^/2=14^2.
(2) 9V4-Vl08=&V4-V(27x4)=9V4-3V4=6V4.
(3) 4v/f-3^2V=^^/15-f,/f=|v'15^i^/15=^yl5.
(4) Add together 3/24, 3^32, and ^192. Here '^2i=%/{SxB)=2l/B, and 8^32=33/(8 X 4)=6V4, also Vl92=V(64 X 3)=4V3 : hence the sum is 2^y/Z-\-6l/i-\- 4V3=6(V3-fV4).
Examples for Exercise.
(1) v'32+v/T2=.
(2) 3V§-2V5V=.
(3) ^/27a*a;+V3a%=
(4) 2^18-Z^8-\-5^50z
(5) ^o?x-s/^=.
(6) .Ja%^^ah^-l^ab=.
(7) In/I-WI +n/15=.
(8) ^(a2_£2)(a_J)+6^(a+&)=.
(9) 3^-16-5^-4=. (10) 2^/72+V24+aV6a;2=.
56. Multiplication and Division of Surds.— BuLE. Reduce the surds to equivalent ones expressing the same root (54) : then multiply or divide, as required —
(1) V2 X V3=2^ X 3*=2^X 3^=6/(23X 32)=V72.
(2) ^/8xV16=8^Xl6*=2.2^x2.2^=4x2^x2^=4.2*=46/32.
(3) (3V2-6V2)--V3=3v|-5.2^--3^=^/6-5V^^/6-|vi08.
Examples for Exercise.
(1) 2^/3x33/4=.
(2) •V4x7V6xv|=.
(3) (is/3)3=.
(4) (23V2)*=.
(5) V12-T-V24=.
(6) 2V^xv|xV16-T-V32=
(7) (3V2-5V2)-r-V3=.
(9) Va-^Va=- (10) (^V18+^V8)xV2=.
57. Binomial Surds. — In the foregoing examples, every divisor has been a monomial surd: when it is a binomial surd, and of one or other of the forms ^/fl^db^/&, %/cL±:%/b, a multiplier of it may always be found that will render the product rational, so that when a fraction occurs with a denominator of such a form, that den. may be rationalized by multiplying both terms of the fraction by the suitable multiplier. When the form of the binomial surd is ^a±:^b, the rational- izing multiplier at once suggests itself, from the property, that the sum
BINOMIAL SUEDS 43
multiplied by the difference of two quantities, gives the difference of their squares: — it is always either ^a—^b, or ^a-\-sJh, thus: — (n/«H-x/Z>)(n/«— v/Z>)=a— &, and {s/ct—,jh){ja-\-s/h)=a—h.,....{^A'\
But when the binomial consists of two cube roots, the rationalizing multiplier is not so obvious : in this case it is trinomial, and consists of the squares of the two surd terms, and of their product with changed sign. This will be sufficiently seen by inspecting examples (4) and (7) at page 15. As in those examples, x and y may each be anything, we may replace them by %/a and \/b, when we shall obviously have i^s/a—X/b) i^^a^+yab + yb'')=a-b,wadi{^s/a-\-yb){X/a-—yah-\-yb~) = a + b.,,\B\
These expressions [^J and [B] furnish all needful directions for ration- alizing binomial surds of the form ^a±Ls/b, or \/a±L\/h.
,,v 3_ 3(2^5+3^7) _3(2s/5+3V7)_ 3
^ ' 2V5-3V7"'{2V5-3V7)(2V5+3V7)~ 20-63 43^ ^ "^ ^ ''
io\ 2 2(V6-V5)_ 2
,„. 1 _V4-V2_(V4-V2)(2+^2)
^ ^ V4+V2~^/4-v'2" 2
<1) 3^(2v'7-3v^5)=.
(2) (2+^/3)^(3+^3)=.
(3) ^/6-^(^/8+^/3)=.
(4) x-^{a—^Jx)=.,
(6) 2^/5-3^/2)+(2^/2+3^/5) (6) 3^(V5+V3)=.
Examples for Exercise.
(7) 2+(V3-V2): a
(8)
Vx-{-yy
58. To extract the square root of a binomial, one of whose terms is rational, and the other a quadratic surd.
Put «+ s/b=(x-{-yY, and a—^b={x—yf, then a=x^+y^..,[l] Also, (a+N/6) [a-^b], a^-b^ix'^-yj .-. ^(a;'-b)=x;'-y\..[2]. From [1] and [2], by addition and subtraction,
// , /j\ /a+v/(a^-5) , /g-V(a2-5) . ,
x-y=s/{a-s/b)=^ 2 V 2 • • • L*J
Consequently, whenever a^—b happens to be a square, the complex surd ^/{a^sjb) may be expressed by the sum or difference of two simple surds, but not under other circumstances.
59. When in any particular example the above-mentioned condition has place, these general formulae will enable us at once to write down the desired simplified expression for the square root of the binomial. But it is nearly as easy to go through the process by which the general results have been obtained, thus : —
Eequired the square root of 16- V 87. Here the necessary condition is fulfilled, for 16^-87=169=132. Put then 16- V87=(^-2/r...[lJ, and 16+ V87=(a:+2/)-...[2].
44 BINOMIAL SUEDS.
Multiplying these together, 266-87=(a;^— y^ .-. n=a!^-if and adding and subtracting 16=a?^+2/'^
29 1 3 1
.-. 29=2^;^ and 3=2/- .'. a!=^-=-s/6S, and tj=^-=-^/Q.
.•.a;-2/=^/(16-^/87)=|(^/58-^/6). Examples fob Exercise.
(1) ^/(2+^/3)=.
(2) s/(8+v'39)=.
(3) >,/(10-V96)=.
(4) v^(7-2V10)=.
(5) V(8+2v^7)=.
(6) V(42+3V174?)=.
(7) V(6+n/20)=.
60. The following are interesting properties of quadratic surds.
1 . The product of two dissimilar quadratic surds cannot be rational. If possible let »ya+ \/b= a rational quantity =p, then ab=p^.
.-. 6=^=4a, and .-. ^hzJ-^a. a a" a
60 that nja and t^h are really similar surds, having the same surd-factor y/a.
2. A quadratic surd can never be equal to the sum or difference of a quadratic surd and a rational quantity.
If possible, let ,^a=b±.^c.'.a=h^-±^h^c-\-c,'. ±.^c= — -7 — ,
so that the surd ^/c is equal to a rational quantity, which is absurd.
3. In every equation of the form a+s/b=a;+^y, if >^b and Vy are surds, then necessarily, a=a;, and .•.b=y.
For since a-\->^b=x+ ^y .'.s/b=x—a-^ ,^y .• .x—a=0, or x=a, otherwise ^b would be = a rational quantity and a surd; .'. also b=y. If a-^s/b=0, then, necessarily, a=0, and 6=0.
4. If >^(a-^^b)=x-{-^, [x, y being one or both quadratic surds) then also must ^(as/b)=x^y, provided that ^6 be a surd. For since a-\-^b=zx''^-\-^xy-{-y', we must have, equating the rational and surd parts, a=x'^-\-y\ ^b = '^xy .'. a—^b=x'—'ilxy-\-y^={x—yy .'. ^(a--y/b) =x-^y.
Note. — Any two quantities may always be expressed by the sum and difference of two others ; for, assuming this sum and difference to be x-\-y and x—y, we can always arrive at determinate values for x and y. The above, therefore, is only a particular case of a perfectly general theorem. It is from the proof of the property in this single case that the expressions [3], [4] at art. (58) are deduced in all the books ; so that they are limited to the condition of ^/b being a surd. But they are entirely unrestricted, being absolute identities ; and the investigation of them, given at (58), is quite independent of the condition that »yb is a surd. Introducing this condition, however, the process for getting s/{ft±.y/b} may be conducted a little differently; thus, taking the example worked at (59) we may proceed thus : —
Pat V(16+ ^/87)=^/a;4-v'y, one of these, at least, being a surd, then 16+^87=
a;+y+2*y^2/ .-. a;+y=16, and 1sjxy=.^^l .'. 4a^=87
29 3 .-. {x+y)^-ixy=16^-87={x-yf .-. x-y=lB : hence, x=—, y=-.
A A
.'. >/(16-V87)=yf -y/2=i(>/58~V6).
IMAGINARY QUANTITIES. 45
The objection to investigating the formulas [3], [4] upon this plan may he removed by referring, not to the principle 4, above, to prove that ±:^'s/xi/=±jh, but to the fact of the double sign of the radical, whether ^fe be a surd or not.
61. Imaginary Quantities. — All the preceding rules of opera- tion equally apply, whether the surds be real or imaginary ; but in dealing with the latter, it will be convenient to replace the imaginary ^ — 1 by the symbol i ; that is, to put — l=i^ in order to avoid all confusion as to signs. Thus, —
(1) (4+^/-3)(3+^/-5)=(4+^/3^/-l)(3+^/5^/-l)=(4+V3)(3^-^V5)= 12+3V3+4V5+*V15=12-V154-(3v'3+4V5)n/-1.
(2) (2+^/-2)(3-V-4)=(2+V2)(3-2t)=6+3V2-4i-2iV2= 6+2^/2+(3^/2-4)^/-l.
i->+GV2+^V3)V-l.
4+V-2_4+V2 (4+V2)(2-V3)_ ^' 2+^/-3~2+iV3"" 4-3i2
(8+V6+ 2(^2-2^/3)^-1)^7 (5) (a+ 60 (c+ di)—{ac--lcl) + {ad+hc)i. a-\-U {a-\-hi){e—di) _ac-\-hd be— ad.
Note. — If expressions of the form a-\-b^ — l be united together by the additive or subtractive signs, or by both, it is plain that the result of the combination will always be of the same form, namely, A+B^ — }. And from the last two examples, it appears that when such expressions are combined by multiplication or division, or by both, the results are still of the same form.
Examples for Exercise.
(1) (2_^_3)V-7=.
(2) (2_V-5)(3-f V-2)=.
<^) (7+^>/-i)h-(1+2V-1)=. (4) (2+ 3V-2)(3-2V-l)=.
(5) '
N/-S+V-6 (6) ^+^-1
3^/-2-2V-3
(7) (3-2^-4)3=.
(8) V(a±V-l)=-* 6*2. We shall conclude this subject by giving an interesting example of
the use of imaginary quantities in investigating certain properties of real quantities. [As above, i is put for ^ — 1, and .-. i^= — 1.]
To prove that the sum of two squares multiplied by the sum of two squares, always gives the sum of two squares for the product.
Let the factors be a^+P, and a'^ + i'"^. Each of these may be produced from a pair of imaginary factors; for they arise from {a + bi)(a—bi), and (a'-f6'f )(«'—&'«), so that the product of the original factors is the pro- duct of the four factors a-\-hi, a—bi, a'-\-bH, a'—b'i. Now, by actually multiplying the first and third of these, and then the second and fourth,
• The result will be found to be of the same form : hence the square root of that result will be of the same form, and so on. (See foot-note, page 41.)
46 TO SOLVE A QUADEATTC WITH ONE UNKNOWN.
we have the following pair, namely, {aa' —bb^)-{-{ab^ + bay, {aa'—bb^^
{ab' -\-ba')i, of which the product is [aa'—bby-\-(ab^-{-ba'f;
.'. (a^ -{-h'){a'''^ -\-b'~)={aa' — bb'f-\-{aV-\-ba'f, which proves the theorem.
For example: let a=5, 6 = 2, a'=3, 6'=4 : then (5- + ?--) (3^4-4-) = (5.3_2. 4)-+(5.4 + 2. 3)- = 7- + 26-=725. Let now a and b be inter- changed, that is, put a=2, & = 5; the first side remains unaltered, but the second side is =(2. 3-5. 4f +(2. 4 + 5. 3)'^=14- + 23-=725. If a' and b' had been interchanged, we should have got the same results. There are thus two ways in which the product of the sum of two squares may be expressed by the sum of two squares : in the instance here ad- duced,(5'- + 22)(3H4-)=7H26'=142 + 23'.*
63. Quadratic Equations. — A quadratic is an equation into which the square of the unknown quantity enters, and enters in such a way as to be removable only by the application of a special rule for the pur- pose. In simple equations, as has already been seen, the square of the unknown sometimes occurs; but then its elimination could always be effected by transposition, division, or some other of the ordinary opera- tions performed upon simple equations. The general form of a quadratic equation with one unknown quantity is this, namely, ax^- ■\-bx^=c ; if the second term be absent, the form is ax~=^c, which is called a pure quad- ratic, the more general one, ax~ -^bx=^c, being an adjected quadratic. A pure quadratic requires no special rule for its solution; for from ax'=^c,
we at once get x
-Vr
64. The rule for solving an adfected quadratic is deduced from the fol- lowing considerations. We already know that x''-{-2ax-\-a^={x + ay, and that x~—^aa)-\-a^={a)—af whatever be represented by a; and we see that the third term, in the first member of each of these identities, is nothing but the square of half the coefficient of the second term. Con- sequently every expression consisting of two terms of the form x^'+mx, or x"^ — mx, can always be made a complete square by merely adding
(I)
for a third term. Thus the expressions x+Qx, x'^—8x, x^' + Sxy
ar—^x, &c., become in this way converted into the complete squares x'''-\- 6a; + 9=(:» + 3)'; a?--8a; + 16=(a;— 4f ; a;H3^ + (ff=(^ + |)-; x^'—hx^ (^ff={x—^^~, &c. The root of each square consisting of two terms, namely, the root of the first term [or), and half the coeff. of the second term taken with its proper sign. Hence the following rule.
65. To solve a Quadratic with one unknown.— Eule. I. Bring the unknown terms to one side of the equation, and the known terms to the other.
2. If the unknown square have a coefficient, other than unity, divide by that coefficient.
3. Add the square of half the coefficient of the simple unknown to each side of the equation ; the unknown side will then be a complete square.
4. Take now the root of each side, affixing the double sign ±: to that
The property in the text may be more briefly proved as follows : —
QUADRATIC EQUATIONS WITH ONE UNKNOWN.
47
of the known side (50), and the quadratic will thus be reduced to a simple equation.
for a;": in the above example w=4. By this change
[A] becomes y^-\-ay=.h , . . [B]
(5) (a;-12)(;c4-2)=0. By actually multiplying, the equation is
the quadratic
a;2_10a;-24=0, which may be solved as above. But the factors of the first member being given, the equation is satisfied by equating either factor to 0; thus, from
ic— 12=0, we get x==.12, and from a;+2=0, we get a;=— 2, which are the two values of x. Whenever we know the factors of an expression, if that ex- pression be =0, the equation may always be solved by equating each factor to 0, since the whole can become 0 only by one or other of the factors becoming 0.
(6) a;-fV(5:K+10)=8. Trans., V(5;r-fl0)=8— ar. Squaring, bx-^lQ=Qi—lQx-\-x^. Trans,, x^—21x=.—^L Completing the square, o «. /21V «r. . 441 225
Extracting the root,
21 15 21+15 ,„
X — = ±-r- .*. X=. =18,
(1) Given 3«2=12a;+15 to find the values of x.
Trans, and dividing by 3, a;^— 4:c=:5.
Completing the square, by adding 2^ to each side, a^— 4a;+4=9.
Extracting the root (that is, writing for the first side, a;— 2), we have ic— 2=±3
.-. a;=2+3, ora;=2-3, that is, a:=5, or —1.
If we substitute 5 for x in the given equation, there results 75=60+15 ; and if we substitute —1 for x, there results 3=— 12+15; the equation being true for either value of x.
(2) 2a2+4a;-3=ll. Trans., 2x^-\-ix=li. -i.2, x^-\-2x=7. Completing the square,
x^-{-2x-\-l=8. Extracting the root,
a;+l=V8=2V2. .•.ar= -1+2^^2, or -1-2^/2.
(3) 3a;2-14a;+15=0. Trans., 3a:2— 14a;=— 15.
14
^3, ^'-y^=-
■5.
Completing the square.
x^
14 /7N.2
■5=
Extracting the root,
7+2 „ 2
^-3=±3-
(4)
X s/x_B
3 2~""2' Multiply by 6, the L.C.M. of den.,
2x—S^x=9. Put y for >^x, .'. 2f—Zy=z9.
. 2_? _?
• • y 2^~2*
Completing the square,
9 9 9
"2^"^ 16
2^16 16
Extracting the root,
3 9 3 9 „ 3
y--=±-.'.y=.-±-=Z,or--.
.-. a;=y'=9, or -. Note. — Every equation of the form
x^"-\-ax"=.b
[A]
may be solved as a quadratic by putting y
2 -2
X2
<^) l(.
2 -2
or 3.
-^y-^-
Put y for -, then the equation is
y(2/'-i)=2(3'-i).
Dividing by y— 1, it becomes
y(y+i)=o» or "
Hy=2-
Completing the square,
1113
Extracting the root.
]4^3
4 2' .'. x=2y=-l±^S.
(8) ar-4-9ar-2=-20. Tut xr-^=y
.:y'^-9y=-20. Completing the square,
/9\2 81 1
48
QUADRATIC EQUATIONS WITH ONE UNKNOWN.
Extracting the root,
that is, —5=5, or 4 .*. a^=r, or -, a? 6 4
.•.«:=±v/5,or±-
There are thus four values for x, either of which will satisfy the proposed equa- tion, which is in fact of the fourth degree.
/gv CT+a; _ a—x
s/a-\-^{a—x) >/a — v^(a— a;) ^ a^\■x_^ya-\-s/{(l—x) a—x s/a — sj{a—x)' Applying the principle at (38) a_ v/a . "^ _ ^ X i^{a—x)"x^ a—x' Clearing fractions, a^—a'^x=:ax^. Dividing by a and trans., x^-\-ax:=a^. Completing the square,
x'^+ax+~=a^-\-j=-^.
Extracting the root,
iC-f
, a —a±ai>J5
0TX=-{-l±^5).
(10) x^-2x-\-6y/{aP-2x+5)=ll. Add 5 to each side, then (a5_2x+5)+6v'(^2_2;c-|.5)=16, or, putting 2p for x^—2x-]-5, ^^+6^=1 6. Completing the square, y'^-^67/-\-9=z25. Extracting the root,
y+3=±5.-.y=2, or -8 .'.y'^=x^-2x-{-5=i . . . (1). OT f=x^-2x-^5=6i . . . (2). From (1), x^-2x=-l. Completing the square, a:^— 2j;+1=0. Extracting the root, a;— 1=0 .*. ic=l.
From (2), a;2-2a;=59. Completing the square,
a;2-2a; 4-1=60.
Extracting the root, a;-l=±s/60=±2v'15 /. a;=l+2V15. (11) ax^+bx=c.
-r a, x^-\—x=-. a a
Completing the square.
Extracting the root,
b_ / 4ac-f62_^^(4ar+&2)
2a .'. x=-
4a2 2a
•5±N/(4ac4-&") 2a
This is a general formula for the solu- tion of every quadratic equation, a, b, c, being any values whatever.
(12) a;-l
=<^0-
Subtract 1 from each side, then
a:-2=l+-4- Add now - to each side,
X
... a;_2+l=H-?-+i. X sjx X
It is easy to see that each side of this
equation is a complete square, the equa.
being the same as
.*, x=.sJx-\-2f or, putting y for pJx, and trans., f-y=2. Completing the square,
Extracting,
y-^±^--'y=-% or-l
.'. x^y'^=:zi^ or 1.
66. The learner may, perhaps feel disposed to inquire : how is it that the double sign is prefixed to the square root of the known side of the equation, and not to that of the unknown ? The answer is, that it is because one side is unknown that no sign is prefixed : we look to the known side for the complete interpretation of the unknown : this inter- pretation furnishing sign as well as value : in the absence of it, we know nothing about the unknown side, and therefore suppress all prefix to it.
SECOND METHOD OF SOLVING A QUADEATIC.
49
Examples for Exercise.
(1) a;24-21a:=100.
(2) a^-26a;+105=0.
(3) 3a;2+ 52^=256.
(4) 5a;2_i2a;+2=ll.
(5) 16a;2-72a;+17=0.
(6) a:+v'(10a;+6)=9.
(7) x-\-^{5x-10)=8.
(8) ^x+2=^{7+2x). x-{-l x—1 x—l~xT^'
(10) x*-8x^=9.
(11) x^-7x-\-^(x^-7x-\-lS)=i2i.
(12) 3a;2-9a:-4v'(a:=^-3:c+5) + ll=0.
(13) i^x+^y-5fx-\-?^=:H.
(9) n:i_tz±=i.
3a;
fX+1
(15) -v/^-2Va;-a;=0.
(17) V(^+21)+N/(:r+21)=12.
(19) ^/a;'^+v'a;3=6V'a?.
(20) a;*-2a^+a;=6.
(21) ^+i+;,+^=4.
(22) ^2+i+4(.:+i)=-3.
67. Second Method of Solving a Quadratic-— When an
equation, by the necessary preliminary operations, is reduced to a quad- ratic of the common form, the preceding method of solving it, by com- pleting the square, frequently introduces fractions, though none occur in the equation itself. The method now to be explained is free from this objection, and is, therefore, well deserving of the student's attention.
If each side of the general quadratic ax'^-{-bx=c, be multiplied by 4^^, the result may evidently be written thus: — (2axf-\-2b{^aa;)=iac, the first member of which becomes a complete square, without fractions, by adding to it fc^ for the third term: we thus have {^axf + ^b{2aa;)+b'^ 4ac + 6^ or, (2aa?+fe)2=4ac^-&^ .'.^ax-\-h=^(4rac + b^). And in this way may any quadratic be reduced to a simple equation at a single step, and at the same time the introduction of fractions be avoided. The formula expressed in words is as follws: —
Rule II. 1. Double the coefficient of x^ in the proposed quadratic: this will give the coefficient of a? in the reduced simple equation.
2. To the first term of the simple equation thus found, connect, with its own sign, the coef. of x in the quadratic : the first side of the simple equation will then be complete.
3. For the second side, multiply the second side of the quadratic (namely c) by 4 times the coef. of a;^ add the square of the next coef. to the result, and prefix to the two terms thus found the radical sign.
Note.— If the first and second coefficients, or the second and third {b and c), be both divisible by 2 or by 4, the division had better be per- formed ; for though a fraction may thus be introduced into the other term, it will disappear from the resulting simple equation.
This rule merely describes how from <i^^ + &a;=c to derive 2aa? + & = v^(4fltc + 6-). Take for instance, the equation Qoc'^-{-6x=l, then by the rule, or by the above formula, l2x-{-6=^{2i-\-2D)=s/4:9=^±7,
— 5±7 1 ,
1 7
Again: let the equation be -;»•=— 3^=- .'. a;— 3=V(7 + 9)=±4
.-. a;=3±4=7 or —1. Lastly, let the equation be 5a;''-~7a;= — 3.
50 GENERAL THEORY OF QUADRATIC EQUATIONS.
68. General Theory of Quadratic Equations.— In dis- cussing the general theory of equations, it is found convenient to have all the significant terms on one side of the equation, and zero, or 0 on the other side: we shall, therefore, write the general quadratic thus: aaP-{- fc^-f-c=0, where a, b, c, are any positive or negative values whatever. For the solution of this equation the formula is (see ex. 11, p. 48, changing the sign of c)
^=zi±NAL'rl2f) (^)
from which we deduce the following inferences, in the expression for which it will be convenient to use two signs not hitherto employed, namely, the signs of inequality. These are > and <, the former of which placed between two quantities implies that the first is greater than the second, the latter, that the first is less than the second.
If b'^=4:ac, the roots are real and equal, for then x= \-0, or , and .
' ^ ' 2a-~ ' 2a' ^ 2a
6^>4ac, ,, ,, real and unequal, for then b^—iac is positive.
[This condition is always fulfilled when c is negative.]
J-<4ac, ,, ,, imaginary, for then b^—4ac is negative,
consequently, the character of the roots of a quadratic may always be
ascertained without solving the equation. Thus, taking the three
1 7
equationsjust solved, namely, 6x^ + 5j?— 1=0, -;??-— 3;r—-=0, bx^—lx-h
3=0, we see that the first two necessarily fulfil the second condition, c
being negative, but in the third 7- < 4.5.3 .-. the roots are imaginary.
b c
The sum of the two values of a; in (A) is — , and their product is - :
^ a ^ a
we infer, therefore, that when any quadratic is divided by the leading co-
b c eflBcient, and is thus in the form aP-\ — ar-j — =:0, that the coef. of the
a a
second term with its sign changed, is always equal to the sum of the roots, and that the third term, without change of sign, is always equal to the pro- duct of the roots.* We may, therefore, readily frame a quadratic that shall have an assigned pair of roots: thus, let the assigned roots be 3 and 5, then the quadratic to which these roots belong will be a?'-— 8d; + 15=0. Let the roots be —3 and 5, then the quadratic is ic-— Qar— 15 = 0. Let them be — 3 and —5, then the equation is a;--f 8;c-}-15=:0. Generally, let the roots be r, /, then the quadratic to which they belong is o)' — (r+/) a;-\- rr'=0, and as the first member of this is the product {x—r){a!—r'), it
b c follows that every quadratic expression of the form x'^-\--a;-\ — , cra?- +
ma}-\-n, is compounded of two simple factors, found thus. Equate the expression to 0 : find the roots r, / of the equation, then the simple factors are {x — r), (a;—/) .* . ax"' -\-bx-\-c=a{x—r){x—r^).
Note.— From the formation of a square, it is obvious that (x—aY may always be converted into (aj+a)^ by adding 4ax ; and that {x+ay will
* In speaking of the roots of an equation, it is always understood that the leading coefficient of the equation, when all the terms are arranged on one side, is positive.
GENERAL THEORY OF QUADRATIC EQUATIONS. 51
become (a?— rt)- by subtracting Aax; and from this fact we may easily solve an interesting problem, namely: — To find two integral squares whose sum shall be an integral square. The solution is
^<^)H^y
where x is any odd integer whatever. When x is odd, x^ must be odd, and, therefore, a;'-±l must be divisible by 2. Suppose x=l, then we have 72 + 242=252. Suppose x=9, then 9^+402=412. The root of the third square always exceeds that of the second by unity, and the sum of these roots is always the first square.
69. Questions producing Quadratic Equations with one unknown Quantity. (1) A company of travellers engaged a conveyance for SI. 15s.; but at the end of the journey two found themselves without money ; in conse- quence of which each of the others had 10s. more to pay : how many
175 persons were there in company? Let x be the number, then
175
shillings is the fair share of each, and is what those who contributed
x—^
actually paid : hence, by the question,
175 175 r= +10 .-. 175x=175a;-350+10a?J-20a: /. 10rc2-20a;=350 .*. x'^-2x=Z5.
Completing the square, a?'— Sa; + 1=36 .-.a;— 1 = ±6 .-. a:=7, or —5. Consequently there were seven persons. The solution a;=— 5 is, of course, inadmissible : either value of x satisfies the equation, but as the question is such as to restrict the answer to a positive number, the nega- tive solution must, of course, be rejected. And similar remarks apply to the values of x in the next question.
(2) Find a number, such, that if 3 times its square root be taken from
4 5 times its fourth root, the remainder may be -.
o
4 Let X be the number, then 5\/x—B»yx=-, or putting y for /^x,
o
l57/-9f=i .'. 9f-15y=-L .-. Rule II. Page 49, 18y-15=V(-144+225)=V81=±9 15±9 4 1 , 256 1 ,^, ,. o,-, 1
••• ^=-18-=3' '' 3 ••• 2^ -^=-8r' °' 81' *^"' ''' ^^^' ''' 81-
(3) A and B leave the same place, at the same time, to travel a dis- tance of 150 miles. A, by travelling 3 miles an hour faster than J5, completes the distance 8h. 20m. before B ; at what rate per hour did A and B travel ? Suppose B goes x miles an hour, then A goes a;+3 : the
time occupied by A is, therefore, — r: hours, and by B, — hours ; and
by the question,
150 . „, 150 , . 150 . 25 150 . .. 6 1_6 __+8-_ that xs, ^3+y=-> or +25, ^+3=",
clearing fractions, 18:c+a;2+3^=18a:+64 .*. x^-]-3x=5i.
.: Eule II., 2a;+3=>/(216+9)=v^225=±15 .-. x=~' ^ =6, or-9.
Hence B goes 6 miles an hour, and A goes x-^3=9 miles an hour.
£ 2
52 HOMOGENEOUS QUADRATICS.
(1) Two travellers, A and B, start from the same place, at the same time, on a journey of 90 miles. A rides one mile an hour more than B, and arrives at his destination an hour before him : at what rate per hour did each travel?
(2) A wine-merchant sold 7 dozen of sherry and 12 dozen of claret for 50Z. : he sells 3 dozen more of sherry for 10^. than of claret for 61. Required the price per dozen of each.
(3) Divide 20 into two parts, such that the product of the whole and one of the parts may be equal to the square of the other part.
(4) Is it possible to divide 1 4 into two parts, such that their product may be 50?
(5) In a certain right-angled triangle the difference between the base and hypotenuse is 8 inches, and the difference between the perpendicular and hypotenuse is 4 inches : required the three sides.
(6) Divide 1 1 into two parts, so that the sum of the cubes of those parts may be 407.
(7) Twenty work-people, men and women, receive 2?. 8s. for a day's work: the men 24s. and the women, 24s. ; but the men receive Is. each more than the women : how many men were there ?
(8) A and B are employed to dig a trench : A alone digs half, and leaves the other half to B ; the work is finished in 25 hours. They are afterwards employed to dig a similar trench, when, both working together, it is dug in 1 2 hours : in how many hours could each alone dig such a trench ?
(9) What two numbers are those whose sum is 39, and the sum of their cubes 17199?
[Additional questions, solvable by quadratics with one unknown, will be found at page 62.]
70. Simultaneous Equations. When one is a simple Equa- tion and the other a Quadratic. Rule. 1. Solve the simple equation for one of the unknown quantities. 2. Substitute the expression for this unknown in the quadratic, from which that unknown is thus eliminated, and then solve the equation for the other unknown.
^ ' I From the first, x= — - — . Substituting in the second,
.-. 23^2+73^=39 .-. (Rule II. p. 49.) 43/-|-7=^/(312+49)=±19. ...2/==^=3, or -61 ....=1±?^=5, or -9 J. Hence the pairs of values are either x=5, y=S ; or else a:=:— 9j, y= — 6^. If both the equations are quadratics, then the elimination of one of the unknowns will frequently lead to an equation of higher degree than the second, the solution of which would require more advanced principles. But there are two classes of simultaneous quadratics, which may always be successfully treated by the rules already established. They are respec- tively homogeneous quadratics and symmetrical quadratics.
71. Homogeneous Quadratics. — A pair of equations is said to be homogeneous when every unknown term is of the same dimensions, that is, when each term has the same number of simple unknown factors. For instance, the terms ix\ 3^2/', 6ary, lif, are each of three dimensions,
S1MMETRTCAL QUADRATICS. ' 53
because each contains three simple unknown factor?, namely, xxx, xyy, ^^y^ yyV' ^^ homogeneous quadratics each term is of two dimensions.
Rule. ]. For either of the unknowns put an unknown multiple of the other ; that is for x put zy, or for y put zx, then the square of this other will necessarily enter every unknown term. 2. Deduce an expression for this square from each equation ; equate the two expressions, from which all but z will be eliminated, and solve the resulting quadratic in z, and thence deduce the values of x and y.
.-. 832+192=14 .-. (Rule II. p. 49) 62+19=v/(12.14+192)=+23.
—194-23 2 4 1 1
- =-, or -7 .-. 0?=-—-—=% or — .-. a;=±3, or +-^^2,
6 3' 2-32+2^ ' 18
_7 and y=zzx=±.2, or 4-^\/2-
The values of x and y, properly paired, are therefore as follow :-
x=Z\ (x=—Z'\
2^=2 r^ b=-2|- l._ 7
,y=-^>/2
-=-^V2
.4s/2
72. And it may here be noticed that, as there are always two unknown factors in each term of the proposed equations, whatever pair of values is discovered for them, there must always be a second pair arising from simply changing the signs of the former pair. We have spoken above of the values of x and y being properly paired. The student must be care- ful to observe consistency in this respect. Whatever value of z gives x, that same value of z must be employed to get the corresponding y, as above.
73. The above method may always be employed with success in the solution of a pair of homogeneous quadratics, but it is not always the shortest method. Rules, perfectly general in themselves, may often be advantageously departed from in particular instances. The exercise of a little independent thought and ingenuity will often suggest facihtating expedients which cannot be embodied in formal rules. These should in general be resorted to only when the penetration and skill of the alge- braist fail to discover to him easier processes.
74. Symmetrical Quadratics.— An equation is said to be sym- metrical when we may change the places of the unknowns without altering the equation or interfering with the condition expressed by it. Thus, the following are symmetrical equations, namely, x+y=a, ax^—hxy-\-ay'^=c, axy—x—y=h, &c. ; for, although we change every x into y, and every y into X, the equation itself remains substantially unchanged. The first and second of these equations are homogeneous as well as symmetrical. The rule for solving a pair of symmetrical quadratics— which rule, indeed, will often succeed for symmetrical equations of higher degrees — is as follows : —
Rule. Substitute for x and y the sum and difference of two other unknowns ; that is, put it + v for x, and u—v for y, in each equation : the result will be a pair of equations with the new unknowns u and v, which pair will always present themselves in a solvable form.
51
MISCELLANEOUS QUADEATTCS.
(3) a?-\-y^-x-y=U
xy-\-x-\-y=\^. Putting u-\-v=.x, and u—v:=y, and remembering that {u + v) ^-\- {u — vY=^ 2{u^+ir), and that {u-{-v){u—v)=u^—iPj we have
u^+i>^-u = 9 .... [1] w2_^_l_2w=19 .... [2]
2u^-\-u=28 EulelL, 4«+l=V225=±15
.*. M=r, or —4.
[1], v2=9+w-w2=7, or -11 .•.r=:+l or+V-ll
7±1 ,'. x=u-{-v=—~-, or — 4±>/— 11
.:x=i, 3, or _4±^/-ll
y=u—v=-—-, or — 4+)^^— 11
.-. y=3, 4, or -4+ V-11- The values of y need not have been worked out ; because of the symmetry of
the equations, if x=:a, y=b, be one pair of values, then x=b, y=a^ must be another pair ; hence from x=a, a;^6, &c. we deduce y=b, y=^a, &c.
(4) o^+y^+x-\-y=18 xy=6 Putting u-\-v=:x, and u—v=^yy w2+i-2+M= 9 .... [1] w2_-y2 _- 6 _ |-2]
;•. Rule II., 4it+l=N/121=±ll
5 .'. w=-, or -3.
.-. [2], r2=ti2_6 1 or 3.
.•.■»=±2' or±V3
.*. X=:u-\-V=.
5±1
or -3±^/3,
2
that is, a;=2, 3, or — 3±,y/3, and therefore^ y=B, 2, or — 3+/y/3.
In the following miscellaneous examples we shall show how both these last may be worked without Eules.
75. Miscellaneous Examples in Simultaneous Quadratics, dc
(1) aP+y^-x-y=lS [1]
xy+x-]-y=l9 .... [2] 1. Add [1] to twice [2], then a>^-\-2xy-\-y^-^x-\-y=i56 that is, (ar-j-y)2-j-(a;-j-y)=56 .-.Rule II., 2(a:+y)+l=^/225=±15 -1+15
'. x+y= — - — =7,
[A]
2. Subtract twice [2] from [1], then a^—2xy+y'^=:Z{x-{-y)—20=l, or —44. .\x-y=±l, or ±2^-11 • • • [B] Adding and subtracting [A], [B], 2;c=7±l, or -8±2^/-ll 2j/=7=Fl, or _8=F2v/-ll .-. xz=i, 3, or -4±^/-ll y=d, 4, or _4=FV-11 (2) x^-]ry^-\-x-^y=18 .... [1] xy= 6 . . . . [2] Adding twice [2] to [1],
{x-\-yy+{x+y)=ZO ... [A] Subtracting twice [2] from [1] (x-2/)2+(^+y)=6 . . . [B]
From [A], x+y=~ ~ =5, or -6
.•. From [B], {x-y)^=l, or 12
.-. x-y=i±\, or ±2^/3
/. 2a;=5±l, or -6+2^3
2y=5+l, or -6+2^^3
.'.x=d, 2, or — 3±>/3
2^=2, 3, or -3=FV3
(3) ^_2?+i=o. ^ ' y X
a;— 2/-l=0.
Trans, and clearing fractions,
x^—f=—xy .... [1]
a;-y=l .... [2]
Dividing [1] by [2], (See ex. 4, p. 15).
x'^-^xy-\ry'^z=z—xy
.'. x^-\-2xy-\-y'^=0
.-. x-{-y=0 ... [3]
Adding and subtracting [2], [3],
1 1
^=2' 2'=-2-
Or thus, x=y-\-l, from 2nd equa.
.-. from 1st, (y-\.lf-y^+y2^y=0 that is, 32/2+3y+l-f-y2+y=o that is, 42/2+ 42/+ 1=0
or, (22/+ 1)2=0
.•.22/+l=0.-.2/=-i.-.a:=-
2
(4) x-y=lj a;3-2/3=:19.
MISCELLANEOUS QUADRATICS.
55
Cubing the first (p. 15),
a?—'i^—Zxy{x—y)=.l that is, a?—y^—Zxy=l.
Subtracting this from the second,
3^:3^=18 .-. 4xy=24, Also from 1st, oi?—2xy-\-y^=z 1
.'. a;+y=±5, and since a?— y=l, .-. a;=3, or —2 ; y=2, or —3.
(5) ar»-|-3/3_i89, a:'y+a;y2=l80. Adding 3 times second to first,
x^+f-\-Zxy{x-iry)=n^.
that is (p. 15), (a:+2/)'=729
/. ;r+y=9 .-. (a:+y)2=81 .... [1]
But since {x-{-y)xy=:lS() .'. a:y=20
.-. ixy=80 .... [2]
Subt. [2] from [1], {x-y^-l . . [3]
.-. [1], [3], x-^y=9, x-y=±l
.*. x=5j or 4 ; y=4, or 5.
(6) 2x^-7 f=-l, xy=Q.
Put a;=«y .-. SvY- 7y^= - 1, ry2=6
"^ 7-3i^= v' .-. 36i;+l=V3025=±55 .-. ?>=■ "
'.^=^,0T
36 .-...,= ^-=±2,
or ±v/-y. .-. x=vy=±2, or if — ^_y- (7) a;+y=a, a;6+y>=&.
Put a:=w+v,y=w— V, then w=-. 2
a*=w'>+5M*i;+10ttV+10wV+5«?;4-f^*
.'. 10%v*+20wV=J-2%5 lUw
.-.«*+ 2mV+%4:
5-}-8u5
lOw ••^=-"'=tv/-10^
But M=-,
a , a , / f a^ . /46+a5)
(8) Find a pair of values for x and y that will satisfy the equations
iC3/4+y=4 From the first, xY='^^-y^.
(4— y)2 From the second, x^y^=. r—
.-. 3/6-103/H3/2-82/+16=0, or, (/_2/4-3/')-8(/+3^-2)=0 which may be put in the form
(3/'-#-8{(/-l) + (y-l)}=0 or, f{f-\Y-8{{jy^-l) + {y-\)}=Q.
This is evidently divisible by y—l; that is, y— 1 is one of the factors of the ex- pression on the left; hence that expression becomes 0 for y— 1=0; in other words, y=l satisfies the equation,
Solutions may often be obtained by thus decomposing the expression equated to 0 into parts, each of which are seen to be divisible by a known factor.
Adding 2 to each side of the first,
\y^x/ Ay^x) 50' ^ 3 203
^-r=-5o-
• Equations of this form are called reciprocal equations, because the roots, -, are the
reciprocals of each other ; that is, if r be one root, then - must be another. If, as in ex. 10, the coeflficients of the terms, when all arranged on one side, are the same when read in order from right to left, as when read from left to right, the equation is always
/ 25 \ a reciprocal equation. Thus, the equation referred to is a;*— 2r'+^2— ^^^a;^— 2a;+
(25 \ ^~144y' ^^' ^^'
56
MISCELLANEOUS QUADRATICS.
o ^_ //40^,9\ /1849_ 43 • •^'"2- V V-25"+iy=V Too— -10
29 7 ic 2/ 29 7
Hence, the first of the given equations is either
x^ .f 87 . 103 641
«2"^a:2 20~^'50 100
y
or -5+
[A], [B].
103_21_ 6_
ic2~60 10~ 100 Subtracting 2 from each side of [A],
(X y\2 441 ^ ic y
100
'2/ X *10'
But^+2'=??. 3^ « 10
a5_6 .*. (Second equa.), 4a;
2 or-.-
*=2^' or -y.
3^=2, or 4x— 5?/=-y
5^=102/— 5y=l0 8
5y=10
60 5 ^ 2 20
...3,=-_...x=-2/=5, orx=-y=--.
And proceeding in a similar way, an- other pair of values may be got from the equation [B].
(10) 12(ic-l)3^=5« y^-x^=.\. From the second, 2'='s/(*^+l)- Substituting in the first,
12(x-l)V(x2+l)=5x. Note. — When dealing with a
Squaring, 144(a;-l)V+l)=25a;2, that is, multiplying out,
144(x4-2a;3+2x2-2x+l)=25x2. Dividing by 144x2^ we have
Adding 2 to each side,
/ IV «/ 1\ 25 .
Putting 2 for xH — , and completing the
square,
---^-^^
13
■1=±12
1 25
25 1
'=12' '' -12
'•*+~=T7>» O'^ *+"=— T^
12
12
25
Completing squares,
, 25 , /25V 49 ^-12^+V24>>=2r2
^12 ^242 242'
1 3
From the first of these, a;=l-, or -.
^ 12'a;-
-=1|, or -\\.
pair
The other values are imaginary. of simultaneous equations, and
having obtained a value of one of the unknowns, we substitute it in one of the given equations, and thence deduce values for the other unknown, it will not always happen that each of these values, taken in conjunction with that substituted, will satisfy the other given equation. As an easy exemplification of this, suppose we havea;— 2/ = l, and ar—y^=S ; then, by division, we get ;»+y=3, from which, by subtraction, we get y=l. Sup- pose, now, we substitute this value of y in the second equation, we then get a;=±2; but the values x=—2, y=l, fail to satisfy the first equation.
76. The student must bear in mind that the sign of a radical is ambiguous only when there is no overruling condition to control it. In ex. 10, for instance, worked out above, if we had substituted the values of a, namely, a:=l^, and a?=|, in the second of the given equations, we should have got y=^{x^ + l)z=+l^, or ±1|, but the first equation ren- ders it necessary that the minus sign be rejected in the first value of y, and the plus sign in the second value.
It is proper, therefore, at least as regards real values, to verify our results by actual substitution in both equations, when such ambiguities occur.
The last term, 1, is with propriety called a coefficient, since the term may be written W, or afi. * See foot note, p. 55.
MISCELLANEOUS QUADRATICS.
67
(1) (x'^-\-xy=12
V (2)
j 2x^-3x7/-\-y"=4: \aP-2xj/-^df=9.
^ ^ \ X7j=Q.
(4) l^'-^=ro^^
^^) \x-\-y=l
(6) .
12.
(8) i^-3'=2
(9) |F"^?=^^ ia;+2/=12.
(10) f ^-2\/^y+y=N/^-N/y
^ ^1
y+-=5.
n2X f(^H/)(^-2')=51
(11)
(13)
(a:+2/)'-27(a:-3/)=0
\{x
^—xy-\-y'^){x—y)=x+y.
if-6i{x-y)= -\-f){x+y)=7Q.
(U)i{x-\-yr-6i{x-y)=0 ^ ^l(x^
77. Miscellaneous Questions requiring Simultaneous Quadratics.
(1) Fiud three numbers, such that their sum may be 33, the sura of their squares 467, and that the difference of the first and second may exceed the difference of the second and third by 6.
(2) Find two numbers, such that their sum, product, and the difference of their squares may all be equal.
(3) What number is that which, if divided by the product of its two digits, the quotient is 2, and, if 27 be added to the number, the digits will be reversed ?
(4) When 962 men were drawn up in two square columns, it was found that one column consisted of 18 ranks more than the other. What was the strength of each column ?
(5) When 732 men were drawn up in column, the number in front and the number of ranks together made up 73. How many ranks were there ?
(6) The fore- wheel of a carriage made 6 revolutions more than the hind- wheel in going 120 yards ; but if the circumference of each wheel were to be increased 1 yard it would make only 4 revolutions more than the hind-wheel in going that distance. Kequired, the circumference of each wheel. ^
(7) Find two numbers, such that the difference of their squares is equal to their product, and the sum of their squares equal to the diffe- rence of their cubes.
(8) A person rows a distance of 20 miles and back in 10 hours, the stream flowing uniformly in the same direction, and he finds that he rows 3 miles with the stream in the same time that he rows 2 miles against it. Required, the time of going and returning, as also the velocity of the stream.
(9) By adding m times the sum of two numbers to the sum of their squares we obtain the square of m, and by adding the product of the numbers to the sum of their squares we obtain the same square m^ : what are the numbers ?
(10) The hypotenuse of a right-angled triangle is to be 13 feet, and
68 THEOBEMS IN PEOPORTION.
the measure of its surface is to be 30 square feet, what must its sides be? [Note. — The surface of a triangle is found by taking half the product of its base and height]
(11) Find the sum of the squares of the roots of the equation ar'-hpx-\-q=0, without solving it.
(12) Find the sum of the cubes of the same roots.
(13) How many sides must a polygon have in order that it may have n diagonals ?
(14) The men in each of two columns of troops A, B, were formed into squares : in the two squares together there were 84 ranks. Upon changing ground A was drawn up with the front that B had, and B with the front that A had, when the number of ranks in the two columns was 91 ; what was the number of men in each column ?
(15) Can two real numbers be foufid, such that their sum, product, and sum of their squares shall all be equal ?
(16) The product of four consecutive positive whole numbers is 3024: find them.
(17) A certain number of two figures is such that the left-hand figure is equal to 3 times the right-hand one, and if 12 be subtracted from the number, the remainder will be equal to the square of the left-hand figure : required the number.
(18) Find two real numbers whose sum is a and product b, and show
that such numbers cannot exist if 6 >(~K ) •
(19) Find two numbers, such that their sum, product, and difference of their squares may all be equal.
(20) There are three whole numbers, such that the squares of the first and second together with their product make 13, the squares of the second and third together with their product make 49, and the squares of the first and third together with their product make 31 : what are the numbers?
78. Proportion. — When one quantity is divided by another of the same kind, the quotient is also called the ratio of the two quantities : the dividend, or numerator of the fraction expressing the ratio, is called the antecedent of that ratio, and the divisor, or denominator of the fraction, the consequent. In expressing the ratio of a to b, which is no other than the quotient a-^b, the small line between the dots is suppressed, and the ratio is written a : b.
Proportion is an equality of ratios ; thus if r = ;^. or which is the same
thing, if a:b=c:d, the four quantities a, b, c, d are said to be in proportion ; but instead of the sign of equality, it is usual to write four dots, thus, a : b : : c : d, which is read as a is to b, so is c to d.
From these explanations it is plain that any two equal fractions may be converted into a proportion, and any proportion may be replaced by two equal fractions. The following are the principal theorems respecting proportion.
79. Theorems in Proportion. — l. The product of the extremes is equal to that of the means, and conversely if two products are equal, the pair of factors from which one is produced will be the extremes, and the pair of factors of the other product, the means of a proportion.
THEOREMS IN PROPORTION. 69
For a\l: '.c \d implies that -=- .■. ad=.lc. 0 d
» • •* » T .1 ad Ic a c ^. ^ . . ,
Again, if ad=hc, then -—=—-.•. -7=-, that is, a:o::c'.d. Id od 0 d
If the two means are equal, that is, if a : b : : b : d, then ad—b'^, that is, if three quantities, a, b, d are in continued proportion, the product of the extremes is equal to the square of the mean. And since also
-=- .-. ( multiplying by - J, -=-=—, so that the first is to the third as
the square of the first to the square of the second, or as the square of the second to that of the third.
2. If four quantities are in proportion, they are also in proportion when
taken inversely, that is, the second is to the first as the fourth to the
third.
a c h d Let a lb: :e : d .'. ,=^ •'• -=- .'.b : a: : d :€, 0 d a c
3. If the quantities be all of the same kind, they are also in proportion when taken alternately, that is, the first is to the third, as the second to the fourth.
Let a : J: :c : d .'. -=- .*. ( mult, by - ), -=- .'. a : c: :6 : d. b d \ c/ c d
Note. — As there cannot be any ratio between two quantities different in kind, alternation can be applied only to quantities all of the same kind.
4. If a : 6: : c :cZ then also a+mb : a-{-nh: :c-\-md : c-\-nd
^ a c a , c , , a , c , *** I" each of these
For since i-=-j •'• T'T^^^j-r^) ^-nd -+71^-4-71. expressions, as well as in
0 d b d b a ^jjg equation below, the
.„ ,. .,. a-^mb C+md . , , ^ , , , . sign ,^ may be put for +.
By dividing, r= ;.'• a-\-mb : a-\-nb: :e-\-md : c+nd.
a-\-rd) c4-nd
Note. — The above is only a particular case of a theorem much more
a c general, for if two fractions are equal as - =-, then we may write any
combinations of a, b, for a new num. and any other combinations for a new den., provided we only take care that each term shall be of the same dimensions as respects a and b, and if in the new fraction thus got we change the a and b for c and d, we shall get another fraction equal to it : 5a^-^ab-\-U^_5c^-^7cd+Z^ * ^^' 6(a+i)^ "" 6{c-]-df
For let VI be a quantity such that c=ma, then necessarily d=mb. Now, if in the second fraction these substitutions be made, and then num. and den. divided by m^ the first fraction will be the result.
5. If any number of quantities of the same kind are proportionals, then as one antecedent is to its consequent, so is the sum of all the ante- cedents to the sum of all the consequents.
heta:b::c:d::e:f, &c. Put ?=3=i, &c. =»i .'• a=mb, c=^md, e=mf, &c. b d f
&+^+/+ &c. '^ .\a:b: -.a+c+ei- &c, : b+d+f+ &c.
.-. a+c+e+ &c. =m(5+c?+/+ &c.) .'. ^^^^^^ ^o='^=b=d ^''
60 VARIATION.
6. If several sets of four be proportionals, then the products of the corresponding antecedents and consequents will be proportional.
mi. ..a c e g h m aeh &c. cam &c.
Thus, if 7=-, 7=T, 7=-, &c., then — "^
h <r f h' I n' ' hflka. dJmkc.
7. If four quantities be proportional according to the algebraic defini- tion, they are proportional according to Euclid's definition.
In order that a, h, c, d may be proportional according to the algebraic
definition, it is necessary and sufiBcient that t=;i. Multiply each by — ,
ma mc
nb nd'
Now, whatever be m and n, if wa>w&, then necessarily mc>wc? ma=:nb, ,, ,, 'mc=:7id ma<C.nh, ,, ,, mc<wd.
And if a, b, c, d, satisfy these conditions, they are proportional according to Euclid.
80. Variation. — This is a term often used instead of proportion, to indicate the same thing, when the quantities compared are variable. We say, for instance, that the price of a commodity varies as its weight, that the work done varies as the time expended on it, the wages paid as the number of men employed, and so on. In all such forms of expression, proportion is of course implied ; but, by aid of the new mark, oc , to stand for varies as, it is more concisely indicated thus : AocB implies that A varies as B, and this means that if A and B be any two corresponding values, and A\ B' any other two corresponding
A B values, then A : A' \\ B : B\ the ratios -j},-^, remaining equal through-
A JO
out all the corresponding changes of A, B.
We know (Arithmetic) that every proportion among magnitudes of
any kind may be converted into a proportion among abstract numbers.
We have only to divide the first two terms by their common unit of
measure, and the other two by their common unit of measure. This
A B A^
being done, we have from the equal ratios — =— , A=^,B, which may
A H Jj
be written A=mB, where m is a constant number, so that AccB may always be changed for A=mB, A and B being the numerical repre- sentatives of the magnitudes or quantities themselves when they are different in kind.
If ^a^, A is said to vary directly as B
liAcc^ „ „ inversely „
If .4 a TV M ,, directly as B and inversely as O ItAcxBO ,, „ conjointly as -B, (7.
We may illustrate all this by a reference to the plane triangle, the sur- face of which, as already noticed (p. 58), is equal to half the product of the base and altitude. Calling the area A, the base B, and the altitude
C, then — 1. If C be invariable, A<xB. 2. If ^ be invariable, Coc ^.
a
ARTTHMETICAL PKOGRESSION. . 61
3. If neither be invariable, B<x—. 4. And AccBC. For 1. A—mB, where m is the constant, \C. 2. C=r-, or BC=m, where m is the con-
stant, ^A. 3. B=m -^, or BC=mA, where m is the constant 2. 4.
A=mBC, where 7?i is the constant ^.
81. Since a variation may always be converted into a proportion, and a proportion into an equation, it follows that problems involving either are virtually problems involving equations only ; but, in certain inquiries, the idea and notation of variation may be very conveniently introduced; as, for instance, in the class of questions which occur in arithmetic under the head of the Double Rule of Three. Take the following for example : —
If 8 men can mow 112 acres of grass in 14 days, how many men will be required to mow 2000 acres in 10 days?
As the number of men varies directly as the number of acres, and inversely as the number of days, we have —
„ . no. of acres 112 2000 men
No. of men oc :; — .". -r-r • -ttt '•'• 8 : no. of men required =200.
no. of days 14 10
The proportion may, of course, be dispensed with; for since
»T /. no. of acres _ ^ ^ . 112 8x14
No. of men =m ~- /.the constant m=:8-7--— -= ,,^'
no. of days 14 112
^^ , 8x14 2000 „^^
.*. No. of men =--—-.— —=200. 11^ 10
82. Arithmetical Progression. — When a set of quantities increase or decrease, each by a constant difference, they are said to be in arithmetical progression. If a represent the first term of the series, d the constant difference — whether positive or negative — and n the number of terms, then the general form of the arithmetical progression is
a, a-\-d, a + 2c?, a-^-M, ,a-Y(n — l)d.
The nth or last term of such a series is usually represented by I, and the sum of the series by S, so that
l=a+{n-V)d, [I.]
and S=a-\-{a-\-d)-\-{a-\-2d) + {a-\-M)-\- . . . . +1
or reversing the terms, -S'=^4- {l—d) + {l—2d) + {l-Zd) + . . . . +a
.-. adding, 2<Sf=(a+0+(a+0 + (a+0+(a+0+ -\-{a+l)=nia-\-t)
.-. -S=4u(a+0, or [I], S=kn{2a+{n-V)d} [IL].
And these two formulae for I and S embrace the whole theory of arithmetical progression.
83. One of the most obvious properties of such a progression is that the sum of the two extreme terms is equal to the sum of any two terms equally distant, one from one extreme, and the other from the other extreme. Let a and I be the extremes, d the common difference, t the term which has m terms before it, and t' that which has m terms after it ; then [I.], t=a-\-md, and l=t' -^-md .: t-l=a^f .'. t-\-f=a + l If t be the middle term, then t=f .*. t=:l{a-^l) that is, any term is equal to half the sum of two terms equally distant from it, for the series may be considered to commence at any term and to end at any term.
62 GEOMETlllCAL PKOGRESSION.
(1) Required, the 16th term of the series 1, 3, 5, 7, ... . Here «=l,w-l = 15, a.ndd='2, .-. ?=1 + J5. 2=31.
(2) Required, the sum of 67 terms of a decreasing arithmetical series of which the first term is 199, and the common difference 3. Here a=199, n-l=66, and d=-3, .-. Z = 199- 66.3=1 .-. ;S=i67.200 = 6700.
(3) It is required to insert four arithmetical means between 5 and 11 ; that is, 5 being the first term, and 11 the sixth, supply the four inter- mediate terms. Hered is at present unknown, but «=5, n=6, and Z=ll; and since [1], ll = 5 + 5rf .'. d=14.; hence the four means are 6|, 7f, 8f, 9|, the entire series being 5, 6|, 7|, 8f, 9|, 11.
(4) How many terms of the series 12, 11|, li, lOj . . . . must be taken to make up the number 55? Here n is unknown, but a=l2, d=-i and S=66 .-. [II.], 66=ln{U-i{n-l)} n 220=?i (49-?i)
.*. w'^— 49w=— 220 .*. n=— ^ — =5, or 44. It thus appears that the
sum of the terms will make 55 whether the number of them be 5 or 44. As the terms continually decrease by |, it is plain that the 25th term must be 0 ; the series then proceeds with negative terms, the first nine- teen of which, balancing the nineteen positive terms before the zero, leaves for the aggregate of the whole 44 terms, only the five leading ones.
Examples for Exercise.
(1) Find the eleventh term of the series 1, 4, 7, 10, &c.
(2) Find the sixth term of the series J, -^^, f , &c.
(3) Find the sum of 7 terms of the series J + 3 + i^ + «"
(4) The first term of an arith. prog-, is 14, and the sum of 8 terms, 28. Required, the common difference.
(5) The sum of the series 1, 5, 9, &c., is 190. How many terms are there ?
(6) Insert five arith. means between ^ and f .
(7) What must the first term of an arith. prog, be whose sum is —28, and the com. difi". of the terms } ?
(8) The first term of an arith. prog, is 3^, the com. diff. 1-J, and the sum 22. Find the number of terms.
84. Geometrical Progression. — A set of quantities are said to be in geometrical progression when each is equal to that which imme- diately precedes, multiplied by a constant quantity, called the common ratio of the progression. According as this common ratio is greater or less than unity is the series increasing or decreasing.
Let a represent the first term, r the common ratio, I the last term, and n the number of terms, then the general form of the geom. prog, is
a, avy ar^, ar^, ar*, ar^, . . ., «r""'=L..[I.].
As the exponents of r form the arithmetical series, 0, 1, 2, 3, 4, 5, 6, 7, 8, &c., and since whatever two of these be taken for extremes, the sum of them will always be equal to the sum of any other two equally distant from them (83), it follows that the product of any two terms of [I.] is equal to the product of any other two terms equally distant from
GEOMETRICAL PROGRESSION. 63
them, as also to the square of the middle term when the number of
terms is odd. Representing the sum of the n terms of the geom. prog.
by S, we have
S=a-\-ar-\-ai^-{-aii^-\-ar*^-\- . . . or"~24-<*»*"~*
,*. <Sr= ar-\-a'fi-\-ar^-\-ar^-{- . . . ar'^~'^-\-ar^~^-\-ar'^
« « « «»*'*—» r"— 1 rl—a
.'. Sr-S=ar''-a .'. S= —=a -= . . . [11.1.
r— 1 r— 1 r— 1
85. The formula [I.] for the last term, and these expressions for the sum, comprehend the entire theory of geometrical progression. When r>\, the terms continually increase in magnitude. If the number of them be indefinitely extended, the sum, therefore, must at length exceed in magnitude all finite limits; but if r<l, and therefore the series be decreasing, the sum always has a finite value, even though the number n of terms be regarded as infinite. For it is plain, when r< 1, that r"," by taking n greater and greater, will at length become less than any number we may assign, however small that number may be. If n be infinitely great, r" must be regarded as absolutely 0, for whatever value we assign to it short of 0 must evidently be too great. In the case, therefore, of an infinite decreasing geom. series, the symbol I, above, must be 0 ; and, putting r for the sum, in this case, instead of S, we have the following expression for the sum of an infinite decreasing geom. prog. :—
.=j^...pii.]
which, upon comparing with [II.], justifies the conclusion that the finding the sura of the terms continued to infinity is really an easier matter than the finding the sum of only a few of the leading terms.
(!) Required the sum of five terms of 1—2+4— .... Here a=l, r=— 2, n=5
... i=(-2)*=16 .-. g^-2-i6-i^n.
— o
111 11^, 1/1\* 1
(2) Sum six terms of g+g+jg ^®^® *=3' *'=2' ^ *** 3\2/ ~96
96
>_Wff-ijTT-#_21_l
■ ^ -3 96 32
1.1 . „ . 1
(3) Required the sum of the infinite series 1+q+q+ • • • • ^^^^ °'^^' ''^3' ^~^
1 3
(4) Find a geometrical mean between - and — . Here ^/{l^i2/6' *^® ^^^^'
12 1 _K 7—?—^ 4
(5) Insert three geom. means between - and -. Here »=2» '*— ^» q~~^ '
. yi^z . »2_? .. ^—j -=-J6 : hence the means are 9 3 ^3 3^
1/211/2 1 /« 1 1 /fi 2V 3' 3' 3V 3' '' 6^^' 3' 9^^* * (6) To find the value of the decimal NFPP . . ., in which PPP . . . represents an infinite series of recurring figures, and N any number of figures preceding the recurring
64 HAEMONTC PROGRESSION.
decimals. Let N consist of n figures, and P oi p figures, then the value of the pro- posed decimal will be
N / P P P ^ \
Where it is plain that the terms within the brackets form an infinite geometric series, of which the common ratio is .
The sum of this series is therefore 2=- * ^ ■* ^
10
n+p
' \ ioV~
iV(lO^-l)+P .
10 (lO^-l)
.*. 'NPPP . . . = — ^i , where 10 —1 is p nines.
lO'^lO^-l)
Hence the fraction, equivalent to the proposed decimal, has for num. iV times the number formed by^ nines, plus P; and for den., that same number of nines followed by n ciphers.
In multiplying by nines, we of course merely annex so many zeros to the number multiplied, and then subtract that number.
Ex. Eequired the fraction equal to •54123123....
Here iV=54, w=2, P=123, ^=3. Also 54 x 999=54000-54=53946. . .54123123 _53946+123_54069^18023
99900 99900 33300*
[This process may be conveniently described in words, as in Weales Rudimentary Arithmetic, p. 138.]
Examples for Exercise.
(1) Find the tenth term of 7, 10, 13, &c.
(2) Find the sum of six terms of 5, 15, 45, &c.
(3) The first term is 7, and the common ratio ^, find 35.
(4) Find the sum of 2| — ^+^— .... to infinity.
(5) Find the sum of the infinite series IH — | — ^+ ....
(6) Insert three geom. means between 2 and 10|-.
(7) Insert five geom. means between 2 and 1458.
(8) Sum the infinite series 2— ^+f — ....
86. Harmomc Progression. — Quantities are said to be in harmonic progression when the reciprocals of them are in arithmetic pro- gression. The designation arises from the fact that musical strings of equal thickness and tension, in order to produce harmony, when sounded together, must have their lengths as the reciprocals of the arithmetical series 1, 2, 3, &c., that is, as ], |^, ^, &c.
If three quantities are in harm. prog, the first is to the third as the difference of the first and second to the difference of the second and
third. For if a, h, c, be in harm, prog., then -, -, -, are in arith. prog.
a 0 c X o
.*. 7 = — 7 .'. mult, by dbc, ac—bc=ab—ac ,-. a : c: :a^h : h—c.
bach J ^
And if a similar property hold with respect to four quantities a, b, c, d, that is, if a : d :: a—b : c—d, the four are said to be in harmonic pro- portion.
HARMONIC PROGRESSION.
65
From the first property, b= is the har. mean between a and c, and
(I ~f"C
ah
^a-h
is a third har. proportional to a and h.
There is no general formula for the sum of any number of terras of an harmonic progression. But as, by taking the reciprocals of the terms, we change the harmonic progression into an arith. prog, many particulars respecting such series may be easily deduced : thus, —
(1) Add two more terms to the harmonic series 1, |, f . The reciprocals are 1, |, |, an arith. prog., in which the com, diff. is — ^ .*. two additional terms are f, §, that is, ^, ^ : hence the required harmonicals are 2 and 3.
(2) Insert three har. means between 2 and 3. Here we have to insert three arith, means between \ and ^. Since a=^, w=5, and 1=^ .*. ^=|+4cZ .*. c?=— ^^ .*. the
-.1, 11 10 9 ,
arith. means are — -, — , — , and
Atc Ai Jti.
'. the har. means are
24 24 24 11' 10' 9 '
or 2^
2f, 2|
Examples for Exercise.
(1) Insert two har. means between 3 and 12.
(2) Insert three har. means between 4 and If.
(3) The arith. mean between two numbers is 2, the har. mean l^ : find the numbers.
(4) The sum of three numbers in har. prog, is 13, and the sum of their squares 61 : what are the numbers ?
87. Miscellaneous Questions in
(1) British wine at 5s. per gal. is mixed with spirits at lis. per gal. in such proportion that the mixture at 9s. per gal. may pro- duce a profit of 35 per cent., what is the proportion ?
Let the prop, of the wine to the spirits be X : y, then 5x-\-lly=: cost of x+y gal. and 9x-\- 9y= selling price, /. Ax— 2y= profit. Hence by the question,
5x-\-lly : ix-2y: :100 : 35 or 20 : 7. Multijjlying extremes and means, and equating the results (79),
••. S0x—4:0y=Z5x+*l7y .•. 45a;=117y .'. 5x=:ldy. Consequently (79) a; : y: :13 : 6, so that the mixture must be at the rate of 13 gal. of wine to 5 gah of spirits.
(2) In a mixture of brandy and water, it was found that if there had been 6 gal. more of each there would have been 7 gal. of brandy to every 6 gal. of water; but if there had been 6 gal. less of each, there would have been 6 gal. of
Proportion and Progressions.
brandy to every 5 gal. of water, what was the original proportion? Let 7x — 6 represent the number of gal. of brandy, and 6x—6 the number of gal. of water, then the first condition will be at once fulfilled; and for the second we have
7a;-12: 6a;-12: :6 :5 .-.(p. 59), X : 6x-12: :1 : 5
.-. 6a:— 12=5a; .*. xz=12 .'. 7a;— 6=78 gal. of brandy Gx—6=66 gal. of water.
(3) The product of three posi- tive whole numbers in geom. prog, is 64, and the sum of their cubes is 584, required the numbers.
Let them be x, xy, a;/, then c(^y^=zQit
and x^-\-xY+x'f=6U.
, 64 ^ 4096 From the 1st, f=—^ .'. f=-^
,'. by substitution in 2nd,
^3+64+i^'=584
... a;8-620a;3=:-4096
.-. a.-3=8 .-. x=2 .: y=2
.'. the numbers are 2, 4, and 8.
66 QUESTIONS IN PROPORTION AND PROGRESSION.
(4) Given the sum S, of n terms of an arith. prog., a-\-b-{-c-\-...-\-l, in which the common difference is S, to find the sum S.^, of n terms of the series of squares a'^-\-b'-rc'^ + ...-\-l^.
Since h=a-\-^, c=b-\-^, d=c-)r^, . . . ., Z=i-j-S, we have
c3=63+3&-^S+3iB2+§^
Adding all these n equations together, omitting the quantities common to both sides, we have
Suppose the arith, prog, to be 1+2+3+...+^, where a=l, and 5=1, also l=n, then
Si=:^n{n-\-l), and by substitution in S.^, we have 8^=- -^ ^ — , that is,
o 2 o
_n^-\-SnP+S7i n(n-}-l) n_n^-\-Zn^-\-2n n(n-\-l) ^"" 3 ~ 2 ~3~ 3 2 •
But the numerator, n{n^-\-S7i-\-2)=n{n-^l){7i+2). [See art. 68.] So that
_2n-l 2~ 6 '
S2=7i{n+1)^~ -J, and since ~ -=
.-. S„ or lH22+32+...+^2=^Vtlp±l).
o
This formula will be found of useful application hereafter (p. 68).
If, instead of the cubes, we take the fourth powers of b,c,d, &c., we may, in like manner, find S^ in terms of /Sg, S ^; and similarly, we may find S^, Sgy &c., each in terms of the sums of the inferior powers.
{\) A starts from a certain place, and travels so that his second day's journey shall be twice the first, the third three times, and so on. B starts 4 days after to overtake him, and travels uniformly per day 9 times the distance A went the first day : when will B come up with A ?
(2) A hundred stones are placed in a straight line, at intervals of 1 yard : how far must a person walk who shall bring the stones, one by one, to a basket, at the place of starting, 10 yards from the first stone?
(3) If the first term of an arith. prog, be n^—n-{-l, and the common difference 2, prove that the sum of n terms will be n'\
(4) Show that the series of cubes V, 2■^ 3'^, 4"\ &c., are severally ob- tained by summing the following portions of the arithmetical series of odd numbers, namely 1 | 3, 5 | 7, 9, 11 | 13, 15, 17, 19 | &c.
(5) A brigade of sappers ' carried on 15 yards of sap the first night, 13 the second, and so on, decreasing 2 yards each night, till 3 yards only were left for the last night. How many nights were they employed, and what was the whole length of the sap ?
(6) A servant agrees to serve his master for 12 years on condition that he is paid a farthing for the first year, a penny for the second, fourpence for the third, and so on : what would be due to him at the end of the 12 years?
PILING OF SHOT— TRIANGULAR PILE. 67
(7) What fraction is equivalent to the recurring decimal -27543543 . . . . ?
(8) Two vessels A and B each contain a mixture of wine and water. In A the wine : water : : 1 : 3, but in B as 3 : 5 : how much must be taken from each vessel to make 14 gal. of mixture containing 5 gal. of wine and 9 gal. of water ?
(9) If a-}-b'(X a—h, prove that a^A-b^ a ab.
10 If the sum of three numbers in arith. prog, be 9, and the sum of their cubes 153 ; what are the numbers?
(11) Find three numbers in geom. prog, such that their sum may be 7, and the sum of their squares 21.
(12) Find four numbers in geom. prog, such that the sum of the means may be 36, and the sum of the extremes 84.
(13) What two numbers are those whose difference, sum, and product are as the numbers 2, 3, and 5 respectively ?
(14) The three digits of a certain number are in arith. prog. : if the number be divided by the sum of the digits, the quotient will be 26 ; but if 1 98 be added to the number, the digits will occur in reverse order : what is the number ?
(15) There are three infinite series, namely : —
^x=l+;+^+...., 5,=!-^^+^-...., ^3=1+1+^+.... Prove that S^xS^^^zS^.
88. Piling of Balls and Shells. — Cannon shot and shells are deposited in arsenals in piles of three different forms, called respectively triangular, square, and rectangular or oblong piles, according to the figure of the base of the pile. Each side of a pile presents an inclined face of rows one above another, each row diminishing by a single shot. If the pile be triangular, or square, it terminates in a single shot at top ; if it be rectangular, it terminates in a ridge, or row of shot. The three different forms of a pile are here represented.
4^ ^
Triangular Pile. Square Pile. Rectangular Pile.
Every horizontal layer of shot is called a course.
89. Triangular Pile.— In this pile each layer or course of shot,
till the top shot is reached, is an equilateral triangle : the side of the first
of these triangles, proceeding from the top, is 2 . * . the number of shot in
that course is 2 + 1 , or 3 ; the number in the next course, the side of which
is 3, is 3 + 2 + 1, or 6 ; the number in the next, the side of which is 4, is
4 + 3 + 2 + 1, or 10; and so on .-. the number of shot in the nth course,
counting from the top is 1+2 + 3 + +?i=in(n + l) (art. 82), so that
if the nth be the bottom course, and we represent the sum of all the
courses by S, we shall have —
n(n-{-l) ,S'=l + 3+6+10+ .... + „ .
(n—l)n , ot(?i+1) or, S= 1 + 3+6+10+ . . . +—5— + — n—
F 2
68 SQUARE PILE — RECTANGULAR PILE.
.-. adding, 25=1+22+32+42+ .... +%2^^:^^+l>
that is (p. ee), 2.='^^!±1|!^)
= "6 . . S= g .... [I.].
If the pile be reduced loj the removal of any of the top courses, the number of shot in the remaining incomplete pile is found by subtracting the pile removed from the original pile.
90. Square Pile. — As the courses here form the series of squares
1+22^32 + ... +,^2^ the number S of shot is ^^K^+l)(2/i + l) |. ^^^^
and the number in an incomplete pile is found from subtracting the pile wanting from the complete pile.
Rectang^ular Pile. — It is convenient to conceive a rectangular pile to be formed thus. To one of the slant faces of a square pile, each side of whose base is equal to the shorter side of the intended rectangular pile, imagine an equal face of shot to be applied (see third /ig. at p. 67) : we shall thus have a rectangular pile terminating in a ridge of two shot, and the base of the rectangle will have one shot more in its length than in its breadth ; and continuing to add faces of shot in this way, when we have added m faces, there must be m + 1 shot in the top ridge, and m shot more in the length of the base than in the breadth. Calling, therefore, the number of shot in the shorter side of the base n, the entire number in the rectangular pile will be —
No. in square pile of base n^, +No. in the m faces added to it.
The number of shot in the m faces will, of course, be m (1+2 + 3 +
n(n + l) , , , . , .1 . n{n + l)(2?i + l) ...+»)=»i-^— — •, and the number in the square pile is ^ -,
.-. S=-^ — ■ ] 3w + 2n + 1 /- . . . [III.], where n is the number of shot in
the shorter side of the base, and m is the difference of the numbers in the longer and shorter sides, or it is the number of shot in the top ridge minus 1. If p represent the number of shot in the longer side of the base, then m=p^n; and making this substitution, 8 becomes
^^!!(!!±i)2^i:^L±i). . . [IV.].
As the number of shot in the top row is m + l=p— /i + l, if we call
1 1, 1 CI n(?i + l)(3( + 2.n — 1) ^-Tn this number t, we shall have 0= -^^ —^ ^'...LV.J.
91. Now, it is worthy of notice that the last factor in the num. of this expression being the same as i + 2(i + 7i— 1), and that p=t-^n — 1, the factor referred to is t-r^p, where t is the top row, and 2p the two equal
base rows parallel to it. Moreover, as -^r — expresses the number of
shot in the triangular or end face of the pile, if we call this number A ,
we may express 8 thus : 8= A — 5-^, which, being independent of n,
o
GENERAL RULE. 69
equally applies to a square pile, in which t—l. With slight modification, it also applies to a triangular pile, for in this, the last factor in the num. of the expression for S is the same as 1+n + l, which we may regard as expressing the top shot + the two opposite base rows, one of these rows being n, and the opposite one merely a single shot. Consequently, allow- ing this latitude to the word row, and representing the two opposite rows of the base, which are parallel to the top row, hyp, p\ we have in general,
for each pile, 8= A ...[VI.], a formula which may be expressed
o
in words as follows : —
92. General Rule.— Multiply the number of shot in a triangular face of the pile by one-third of the number of shot in the three parallel edges of the pile : the product will be the number in the pile.
In all the cases an incomplete pile is regarded as the difference of two complete piles.
(1) How many shot are there in an incomplete triangular pile of 1 5 courses, 6 shot being in a side of the upper course ? As 6 shot are in the side of the upper course, 5 courses have been removed, so that the com- plete pile had 20 courses : hence in the formula [I.] putting first 20 for n^
and then 5, and subtracting, we have iS= — '- — '—^ '—^ =7(10.22—5)=