The New International Encyclopædia/Division in Mathematics

DIVISION. In mathematics, one of the four fundamental processes of arithmetic, the one by which we find one of two factors when the product and the other factors are given. The given factor is called the divisor, the given product is called the dividend, and the result (i.e. the required factor) is called the quotient. The definition of division leads to the following identity: dividend = divisor × quotient + remainder. If the remainder is zero, the division is said to be exact. The common symbols for division are: a/b, a ÷ b; a : b; a/b, ab^-1, in which a is the dividend and b the divisor. Two forms of division are recognized in elementary arithmetic, the one based on the idea of measurement and the other on the idea of partition. The fonner is the case of dividing one number by another of the same kind, and the latter that of dividing a concrete by an abstract number.

The usual tests of the correctness of division are: (a) multiply the quotient by the divisor and add the remainder, the result equaling the dividend; (b) compare the excesses of nines in the identity of division. See Checking.

Simple tests of the divisibility of numbers by 2, 4, 5, 6, 8, 9, 10, 11 are: (a) a number is divisible by 2, 4, or 8 if the number represented by the last digit, the last two digits, or the last three digits is divisible liy 2, 4, or 8 respectively; (b) a number is divisible by 5 if it ends in 0 or 5, by 10 if it ends in 0; (c) a number is divisible by 9, or by 3, if the sum of its digits is divisible by 9, or by 3, respectively, and by 6 if it is even, and the sum of its digits is divisible by 3; (d) a number is divisible by 11 if the difference between the sum of the digits in the odd and in the even places is divisible 11. The simplest test of the divisibility by algebraic binomials is that of the remainder theorem (q.v.). The division of large numbers is generally facilitated by the use of logarithms (q.v.). For the origin of the present method of division and for improved forms, see Arithmetic.