immaterial, the product is indifferently written abc, acb, bac, bca, cab, cba.

20. Def. In an expression such as abc, denoting the multiplication of two or more quantities together, any one of these is called the coefficient of the product of the others. Then a would be called the coefficient of b in ab, and of bc in abc, c the coefficient of ab in abc. Unity is the coefficient of a standing alone, though it is not usual to write 1a, but the letter a only.

21. Def. Quantities multiplied together are called factors of the result. Thus a, b, c, are three factors of the quantity abc, which is produced by multiplying them together.

LAW OF SIGNS IN MULTIPLICATION.

22. When two positive quantities, as a and b, are multiplied together, the product means the result of taking the multiple, or part of one designated by the other, and therefore is positive, since nothing in this process has altered the character or affection of either of the two quantities.

If a negative quantity, -a, is multiplied by a positive quantity, b, the multiple or part so taken of a is negative still, and thus the product is negative.

The negative sign having a power of reversing the character or affection of the magnitude to which it is prefixed, multiplying by a negative multiplier means the taking the multiple or part thus designated, and reversing besides the sign of the quantity multiplied. Hence:

1. If a positive quantity, a, is multiplied by a negative quantity, -6, the multiple or part expressed by b is taken, and the sign of a is reversed as well, whereby the result is -ba, or -ab.

2. If a negative quantity, -a, is multiplied by a negative quantity,b, by the same reversal of sign the result is ba or ab.

These results are of the highest importance, and may be

thus placed in a tabular form. In the multiplication of two algebraic quantities

results which are expressed in words in the rule, that in multiplication 'like signs give + and unlike signs give -.' 23. Conversely, the fact of the product of two quantities being positive certifies that these quantities are either both. positive or both negative, and the product being negative certifies that the quantities have different algebraical signs. 7×5 or (-7)x(-5) = 35 7x (-5) or (-7) × (5) = -35.

Thus :

and

13, or

The number 39 can arise as the product of 3 of −3 and −13, while -39 can arise from the multiplication of 3 and -13 or of

3 and 13.

24. In the product ab, if either of the quantities a or b be zero, or have no magnitude, then whatever finite value the other may have, the product is zero, because if a, for instance, has no magnitude it cannot give a result of any magnitude by being taken any defined number of times.

DIVISION OF ALGEBRAIC QUANTITIES.

25. If, in the ordinary arithmetical sense, a quantity a is divided by a quantity b, the result is expressed by writing these letters in the form of a fraction, the divisor being in

α

b

place of denominator, and the dividend in place of

numerator.

In division the law of signs above stated (22) holds

a

good, the result being positive or negative as a and b

have the same or different signs. If, for the sake of descrip

tion, we use the terms dividend, divisor, and quotient, in the usual arithmetical sense, from the idea of division we know that the divisor and quotient multiplied together produce the dividend. Hence :

1. If the dividend is positive,

and

the product of the divisor and quotient is positive, the divisor and quotient have the same sign (23) ; .. if the divisor is positive, so is the quotient, if the divisor is negative, so is the quotient;

2. If the dividend is negative,

the product of the divisor and quotient is negative, the divisor and quotient have opposite signs (23); .. if the divisor is positive, the quotient is negative, if the divisor is negative, the quotient is positive.

To collect these results, when the divisor and dividend. have the same sign, the quotient is positive; when the divisor and dividend have different signs, the quotient is negative.

26. Division is sometimes expressed in writing by the

symbol +, a+b meaning that a is divided by b; though a÷b

a

is the more usual and the more convenient manner of expressing this operation.

ALGEBRAICAL FRACTIONS.

a

27. An expression such as wherein the result of a

division is indicated without being performed, is an alge

braical fraction. When a and b mean positive integers, it has the signification of a vulgar fraction in Arithmetic, meaning that when unity is divided into b equal parts, a of these parts are taken. In every case an algebraical fraction will mean a quantity which, if it be multiplied by b, will b

give the result a.

α

In the fraction, the quantity a is called, as in Arithmetic, the numerator, and b the denominator.

28. If a and b have the same signs the fraction is positive (25), if different signs it is negative. If the sign of either a or bis changed, the sign of the fraction is changed, inasmuch as if they previously had the same, they now have different signs, and the fraction is changed from being positive to be negative. If a and b had previously different signs, they have now the same, and the fraction, from being negative, becomes positive.

29. If the numerator of a fraction be zero while the denominator is finite, the fraction has no magnitude or is For if it had any magnitude, the multiplication of that magnitude by the denominator must produce a result different from zero the numerator.

zero.

If the numerator be finite and the denominator zero, the fraction is then beyond numerical representation. For if it were supposed to have any defined magnitude, this magnitude, multiplying the denominator, could still produce no result but zero, and could not therefore give the numerator.

INVOLUTION.

30. Def. The result of multiplying an algebraic quantity by itself is called its square or second power. Thus, a xa is written a2, and is called the square or second power of a, a itself being, by analogy, called the first power of a.

So the square multiplied by a, or the product axaxa, is called the cube or third power of a, and is written a3.

These terms have been suggested by the geometrical facts that if a be the number of units of length in a line, suppose 4 inches, then 4× 4, or 16, is the number of square inches in the square described on that line, and 4 × 4 × 4, or 64, is the number of cubic inches in the cube of which this line is an edge.

By extending this notation, axaxaxa is written a4, and called the fourth power of a, and generally if there be any number of times m, that the quantity a is repeated in continued multiplication, the result is called the mth power of a, and is written am.

These operations are sometimes termed the raising a to the second, third, fourth, &c., powers, and the number expressing that power is called the index or exponent. Thus, 5 is the index or exponent of a in a5, which is called a raised to the 5th power.

31. When the quantity raised to a power is negative, its square is positive by the law of signs (22), its cube is negative, its fourth power is positive, and generally the power is positive or negative as the exponent is even or odd respectively.

EVOLUTION.

32. Def. A quantity which multiplied by itself gives the result a, is called the square root of a, and is written a. Thus, ax √a=a or a1× a1=a.

a or

Obs.-The symbol is supposed to be a perverted form of the letter r, the initial letter of the word root.

A quantity whose cube or third power is a is called the cube root of a, and is written 3/a, or a1.

1

By extension of this notation / a, or am, means a quantity which, raised to the mth power, produces a.

33. A negative quantity can have no numerical square root, because any numerical quantity, be it positive or