Sixth Edition. Globe 8vo. Price 4s. 6d.

ELEMENTARY ALGEBRA

BY

CHARLES SMITH, M.A.

MASTER OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE

SIXTH EDITion, revised AND ENLARGED

London

MACMILLAN AND CO., LIMITED

NEW YORK: THE MACMILLAN COMPANY

For the Second Edition the whole book was thoroughly revised, and the early chapters remodelled and simplified, and chapters on Logarithms and Scales of Notation were added. The number of examples has been very greatly increased.

CONTENTS: Definitions; Positive and Negative Quantities; Addition; Subtraction; Brackets; Multiplication; Division. Miscellaneous Examples I.: Simple Equations; Problems; Simultaneous Equations of the First Degree; Problems. Miscellaneous Examples II.: Factors; Highest Common Factors; Lowest Common Multiples; Fractions; Equations with Fractions. Miscellaneous Examples III.: Quadratic Equations; Equations of Higher Degree than the Second; Simultaneous Equations of the Second Degree; Problems. Miscellaneous Examples IV.: Miscellaneous Equations; Powers and Roots; Square Root; Indices; Surds; Ratio, Proportion, Variation. Miscellaneous Examples V.: Arithmetical Progression; Geometrical Progression; Harmonical Progression ; Other Simple Series. Miscellaneous Examples VI.: Permutations and Combinations; The Binomial Theorem; Logarithms. Miscellaneous Theorems and Examples: Scales of Notation. Answers to the Examples.

Academy-"Whatever gifts are required for the composition of a good school textbook on a mathematical subject seem to have been bestowed upon, or to have been acquired by, our author. His Conics and his two Algebras are praised by all teachers, and have attained a large circulation in this country, and besides are 'in very general use in schools and colleges throughout the United States."

Athenæum-"This Elementary Algebra treats the subject up to the Binomial Theorem for a positive integral exponent, and so far as it goes deserves the highest commendation. Mr. Smith has avoided the danger, which, as the preface shows, besets writers of treatises like the one before us-that of 'paying too little attention to the groundwork of their subject.' All through the volume the reasoning underlying the processes of algebra is kept prominently in view, and thus a real interest is infused into the subject, while the educational value of the study is immensely increased. This valuable characteristic of the book is observable as much in the earliest as in the most advanced chapters, and we doubt not that beginners will appreciate it. The examples, which are very numerous, are a notable feature of the book, and, so far as we have investigated them, are singularly well selected and arranged, and the solution of them on the students' part, after careful perusal of the chapters to which they are appended, cannot fail to be greatly 'for the benefit of beginners.

Saturday Review "One could hardly desire a better beginning on the subject of which it treats than Mr. Charles Smith's Elementary Algebra. It is instinct with the merits which distinguished his previous ventures, and has the same lucidus ordo. A very carefully selected collection of exercises adds considerably to the work."

...

Nature "It is a pleasure to come across an algebra book which has manifestly not been written in order merely to prepare students to pass an examination. Not that we think Mr. Smith's book unsuitable for this purpose; indeed, with its carefully-worked examples, graduated sets of exercises, and regularly recurring miscellaneous examination papers, it compares favourably with the most approved 'grinders' books. . . . He shows to great advantage as a teacher, his style of exposition being most lucid; the average student ought to find the book easy and pleasant reading. The second set of exercises on the Binomial Theorem is worth specially noting. This is a second edition of this well-known book, and differs from the first in some important particulars. It has been thoroughly revised, and the early chapters have been simplified and remodelled. Chapters on logarithms and scales of notation form a useful and valuable addition, and there is a great increase in the number of the examples. For beginners this work should prove invaluable, and even more advanced students would do well to glance over its pages."

Educational Times-"There is a logical clearness about his expositions and the order of his chapters for which both schoolboys and schoolmasters should be, and will be, very grateful. His treatment of the Theory of Indices, for instance, though really a very simple matter, is admirable for the way in which it sets forth the difficulties of the subject, and then solves them."

With the above definition multiplication by a negative quantity presents no difficulty.

For example, to multiply 4 by 5. Since to subtract 5 by one subtraction is the same as to subtract five units successively,

(-5) × (-4)=-(-5)-(-5)-(— 5) − (− 5)

We can proceed in a similar manner for any other numbers whether integral or fractional, positive or negative. Hence we have the following laws :

The rule by which we determine the signs of the products is called the Law of Signs: this law is sometimes enunciated briefly as follows:-like signs give +, and unlike signs -.

39. The factors of a product may be taken in any order.

It is proved in Arithmetic that when one number, whether integral or fractional, is multiplied by a second, the result is the same as when the second is multiplied by the first.

FACTORS.

EXAMPLES. XXVII.

Find the factors of each of the following expressions:

99

3. 2x2+11x+12.

5. 3x2+7x-6.

6. 4x2+x-3.

8.

3x2+11x - 20.

11.

7x2+75x - 108.

14. 4x2+4x-15.

17. 132x2+x-1.

20. 7x2+123x – 54.

21. 24x2-30x - 75.

24. x2-11xy+18y2.

23. x2-6xy+8y2.

26. x2-25xy+150y2. 27. x2-35xy-200y2. 29. 7x2-33xy – 54y2.

36. 15x4y-4x3y2 – 4x2y3.

31.

x4-13x2+36.

33. 36x4-97x2y2+36y*.
35. x3y - x2y2 - 2xy3.

37. 75xy3-130x2y1 — 9x3у5.

102. It is clear that the method of finding by trial the factors of an expression of the form px2 + qx+r, where P, q, r are known numbers, would be very tedious if there were many pairs of numbers whose product was Ρ and many pairs whose product was r, for there would then be very many pairs of factors which would agree with the given expression so far as the end terms were concerned, and out of these the single pair which would give the correct middle term would have to be sought. It would, for example, be almost impossible to find the factors of 2310x2-2419x-9009 in this way.

Again, we not unfrequently meet with such an expression as x2 + 6x +7 which cannot be written as the product of two factors altogether rational, and in such a case it would be impracticable to try and guess the factors.

We therefore need some method of finding the factors of a quadratic expression which is applicable to all cases. This method we proceed to investigate.

186

SIMULTANEOUS EQUATIONS OF

157. It should be remarked that we cannot solve any two equations which are both of the second degree; for the elimination of one of the unknown quantities will frequently lead to an equation of higher degree than the second, from which the remaining unknown quantity would have to be found; and we cannot solve an equation of higher degree than the second, except in very special cases.

For example, if we have the equations x2+x+y=3 and x2+y2=5. We have from the first equation y=3-x-x2; and, by substituting this value of y in the second equation, we get x2 + (3 − x − x2)2=5,

that is

x4+2x3-4x2-6x+4=0;

and this equation of the fourth degree cannot be solved by any methods which are within the range of this book.

158. We can always solve two equations of the second degree when all the terms which contain the unknown quantities are of the second degree. The method will be seen from the following example.

Ex. 1. Solve the equations

x2+3xy=28,

xy+4y2=8.

Divide the members of the first equation by the corresponding members of the second equation; we then have

I. If x=4y, we have from the second equation,

4y2+4y2=8; .. y = ±1.