A binary operation in mathematics, the inverse operation of multiplication. Its operands are its dividend and its divisor; its result is its quotient.

A word of warning:

A long time ago, the author, quite an enthusiast of E2, sought a way to make a contribution. A great comfort and pleasure in his life was Mathematics; for example he spent many hours reading about the Extraction of Roots in the Encyclopedia Britannica, and the occasional mathematical problem within his means was all the sustenance he needed.

Having seen many attempts at the Rigorization of Mathematics, directed at the Youths in High Schools today, the Foundations seemed important. Certainly they are poorly understood, if the many writeups here on, for example, division by zero are any indication. So the task became the telling of what he knew of division - particularly its DEFINITION, which is not commonly known.

Various concepts of division exist - it is how we check multiplication; it is an idealization of physical partitioning into equal groups; it creates formal symbols for which multiplication by the divisor returns the dividend - in this way we can divide by numbers which are not factors of the dividend... There is a mess of algorithms and notations and definitions, but in the end it seems that division is not very important or interesting. The author is at a loss.

(If there is any interest, he will tell about formal division — how mathematicians narrowly escape dividing by zero —, division viewed as a continuous function, or division into factors as a method for understanding an algebraic structure. Let him know.)

#### Operation

Division is breaking a dividend into parts each equal in magnitude to the divisor and counting them. Equivalently, one can break a dividend into a number of equal parts represented by the divisor and measure each part's size. But perhaps the most precise and simple definition is that division is the inverse operation of multiplication.

(a ÷ b = c) if and only if (c × b = a).

Dividing by a number in a group (such as a nonzero real number), then, is equivalent to multiplying by its multiplicative inverse.

(a × b-1) × ba × (b-1 × b) = a × 1 = a;
applying the definition of division, a/b = ab-1 if the latter is defined.

Addition distributes on the left over division because it distributes over multiplication.

When a/c + b/c = q, (a/c + b/c)c = qc,
and since multiplication is distributive, a + b = qc;
so (a + b)/ c = q.

It is not always possible, however, to distribute (a + b)/c = q; for example, as integers (1 + 3)/2 = 2, but 1 and 3 are not divisible by two.

#### Notation

Division is usually written horizontally with the operator in the middle. There are at least three such ways to write a quotient:

• As a fraction, a divided by b is written with a above and b below a horizontal bar. By analogy, fractions are often written diagonally or horizontally, as a/b.
• As a ratio, a divided by b is written horizontally, as a : b.
• The generic symbol for division is the division sign ÷, a colon superimposed on a horizontal bar, which reads as "divided by" instead of "over" or "to".

In long division, the long division symbol or the colon is used to separate the divisor from the dividend.

#### Algorithms

The division algorithm (see abiessu's writeup there) allows splitting division problems into smaller problems.

People usually divide small two digit numbers by one digit numbers in their heads using shortcuts and multiplication tables. If all else fails, you can guess and check.

#### Multiplication by the multiplicative inverse

To divide fractions, you may multiply the dividend by the reciprocal of the divisor. To divide integers modulo n, multiply the dividend by the multiplicative inverse of the divisor. In other cases, another method is probably simpler.