Division

Division

Division is the inverse of multiplication; if a × b = c, then a = c ÷ b, as long as b is not zero. Division by zero is undefined for the real numbers and most other contexts,^:because if b = 0, then a cannot be deduced from b and c, as then c will always equal zero regardless of a. In some contexts, division by zero can be defined although to a limited extent, and limits involving division of a real number as it approaches zero are defined.

In division, the dividend is divided by the divisor to get a quotient. In the above example, 20 is the dividend, five is the divisor, and four is the quotient. In some cases, the divisor may not be contained fully by the dividend; for example, 10 ÷ 3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, so 10 ÷ 3 is equal to 31/3 or 3.33 . . ., but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded.

Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called.

Video: Math Basic Division

Properties

Division is right-distributive over addition and subtraction. That means:

in the same way as in multiplication . But division is not left-distributive, i.e. we have

unlike multiplication.

If there are multiple divisions in a row the order of operation goes from left to right, which is called left-associative:

.

Division of integers

Division of integers is not closed. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches:

1. Say that 26 cannot be divided by 11; division becomes a partial function.
2. Give an approximate answer as a decimal fraction or a mixed number, so  or This is the approach usually taken in numerical computation.
3. Give the answer as a fraction representing a rational number, so the result of the division of 26 by 11 is But, usually, the resulting fraction should be simplified: the result of the division of 52 by 22 is also . This simplification may be done by factoring out the greatest common divisor.
4. Give the answer as an integer quotient and a remainder, so To make the distinction with the previous case, this division, with two integers as result, is sometimes called Euclidean division, because it is the basis of the Euclidean algorithm.
5. Give the integer quotient as the answer, so This is sometimes called integer division.

Video: Dividing Integers

Division of rational numbers

The result of dividing two rational numbers ......

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