SECTION IV.

MULTIPLICATION.

(53.) MULTIPLICATION is repeating the multiplicand as many times as there are units in the multiplier.

CASE I.

When both the factors are monomials.

If the quantity a is to be repeated five times, we may write it thus:

a+a+a+a+a,

which is equal to 5a; that is, a multiplied by 5 is equal to 5a.

Ifb is to be repeated six times, we may write it b+b+b+b+b+b,

which is equal to 6b.

If x is to be repeated any number of times, for instance as many times as there are units in a, we may write it ax, which signifies a times x, or x multiplied by a.

Again, if ab is to be repeated four times, we may write it ab+ab+ab+ab,

which is equal to 4ab, or four times the product ab.

(54.) When several quantities are to be multiplied together, the result will be the same in whatever order the multiplication is performed. Thus, let five dots

QUEST. What is multiplication? What is case first? Is it material in what order the multiplication be perfcrmed?

be arranged upon a horizontal line, and let there be formed four such parallel lines.

Then it is plain that the number of units in the table is equal to the five units of the horizontal line repeated as many times as there are units in a vertical column; that is, to the product of 5 by 4. But this sum is also equal to the four units of a vertical line repeated as many times as there are units in a horizontal line; that is, to the product of 4 by 5. Therefore the product of 5 by 4 is equal to the product of 4 by 5. For the same reason, 2×3×4 is equal to 2×4×3, or 4×3×2, or 3×4×2, the product in each case being 24. So, also, if a, b, and c represent any three numbers, we shall have abc equal to bca or cab. It is, however, generally most convenient to arrange the letters in alphabetical order. If a man earn 4x dollars a month, how much will he earn in 5y months?

Here we must repeat 4x dollars as many times as there are units in 5y; hence the product is

which is equal to 20xy.

4xX5y.

(55.) Hence, for the mutiplication of monomials we have the following

RULE.

Multiply the coefficients of the two terms together, and to the product annex all the different letters in

succession.

QUEST.-Give the rule for the multiplication of monomials.

(56.) We have seen, in Art. 27, that when the same letter appears several times as a factor in a product, this is briefly expressed by means of an exponent. Thus aaa is written a3, the number three showing that a enters three times as a factor. Hence, if the same letters are found in two monomials which are to be multiplied together, the expression for the product may be abbreviated by adding the exponents of the same letters. Thus, if we are to multiply a' by a', we find a equivalent to aaa, and a2 to aa. Therefore the product will be aaaaa, which may be written a3, a result which we might have obtained at once by adding together 2 and 3, the exponents of the common letter a.

Hence, since every factor of both multiplier and multiplicand must appear in the product, we have the following

RULE FOR THE EXPONENTS.

Powers of the same quantity may be multiplied by adding their exponents.

QUEST.-Give the rule for the exponents in multiplication.

(57.) When the multiplicand is a polynomiaí.

If a+b is to be multiplied by c, this implies that the sum of the units in a and b is to be repeated c times; that is, the units in b repeated c times must be added to the units in a repeated also c times. Hence we deduce the following

RULE.

Multiply each term of the multiplicand separately by the multiplier, and add together the products.

QUEST.-What is the second case in multiplication? Give the rule.

(58.) When both the factors are polynomials.

If a+b is to be multiplied by c+d, this implies that the quantity a+b is to be repeated as many times as there are units in the sum of c and d; that is, we are to multiply a+b by c and d successively, and add the partial products. Hence we deduce the following

RULE.

Multiply each term of the multiplicand by each term of the multiplier separately, and add together the products.

(59.) When several terms in the product are similar, it is most convenient to set them under each other, and then unite them by the rules for addition.

QUEST.-What is the third case in multiplication? When several terms are similar, how do we proceed?

Give the rule.