I. Let a and b be positive numbers, and c a negative number. Let cc'; then c' is a positive number (Art. 31). Now, a(b+c)= a(bc') ab ac', by (2). = - Also, ab + ac = ab + a( — c') = ab — ac'. (3) In like manner, it may be proved that (1) holds when a and c are positive numbers, and b a negative number. II. Let a be a positive number, and b and c negative numbers. Let bb', and c-c'; then b' and c' are positive numbers. III. Leta be a negative number, and let b and c be either positive or negative. Let aa'; then a' is a positive number. Whence, a(b+c) = ab + ac. Thus (1) is proved to hold for all positive or negative integral or fractional values of a, b, and c. The result expressed in equation (1) is called the Distributive Law for Multiplication. By aid of the Commutative Law (Art. 58), equation (1) may be written in the form 61. It follows from the definitions of Arts. 50 and 53 that 0x a = 0; for if 0 is added any number of times, the result is 0. If we could assume the Commutative Law (Art. 58) to hold with respect to the product. 0 x a, we should have a x0 = 0. A rigorous proof of this result will be given in Chapter XIII. DEFINITIONS. 62. If two or more numbers are multiplied together, each of them, or the product of any number of them, is called a factor of the product. Thus, a, b, c, ab, ac, and bc are factors of the product abc. 63. Any factor of a product is called the Coefficient of the product of the remaining factors. etc. Thus in 2ab, 2 is the coefficient of ab; 2a of b; a of 2b; 64. If one factor of a product is expressed in Arabic numerals, and the other in letters, the former is called the numerical coefficient of the latter. Thus in 2ab, 2 is the numerical coefficient of ab. If no numerical coefficient is expressed, the coefficient unity is understood. Thus, a is the same as 1 a. In a negative term (Art. 19), the numerical coefficient is. understood to include the sign. 65. We shall use the term integral expression to denote a rational and integral expression (Art. 23), with integral numerical coefficients; as 2x2-3 ab+c3. 66. Similar or Like Terms are those which either do not differ at all, or else differ only in their numerical coefficients; as 2ay and -7x2y. Dissimilar or Unlike Terms are those which are not similar; as 3ay and 3xy2. DIVISION. 67. Division, in Arithmetic, may be defined as the process of finding one of two numbers, when their product and the other number are given; and we shall attach this meaning to the operation in all cases where the numbers involved are positive or negative integers, or positive or negative fractions. The Dividend is the product of the numbers. ▾ The Quotient is the required number. The quotient when a is divided by b is expressed 2. Then since the product of the quotient and the divisor is equal to the dividend, we have Multiplying each of these equals by a (Art. 54), we have ах x b = a. 'Since the dividend is the product of the divisor and the quotient, we may regard a as the dividend, b as the divisor, That is, to divide a by b is the same thing as to multiply a by the reciprocal of b (Art. 10). Then by the Commutative and Associative Laws for Multiplication (Arts. 58, 59), That is, if equal numbers are divided by equal numbers, the results are equal. Multiplying each of these equals by c, we have ac= bcx. (1) Regarding ac as the dividend, be as the divisor, and x as the quotient, this may be written From (1) and (2), by Art. 27, ac a = bc b (3) That is, a factor common to the dividend and divisor may be removed, or cancelled. That is, if a number is both multiplied and divided by the same number, the value of the former will not be changed. Since the dividend is the product of the divisor and quotient, these may be written in the forms From these results we may state what is called the Rule of Signs with regard to the quotient of two terms: divided by + divided by +, and Hence, in Division as in Multiplication, Like signs produce +, and unlike signs produce dividend, a as the divisor, and 0 as 0. a |