# Division (mathematics)

In mathematics, the word "**division**" means the operation which is the opposite of multiplication. The symbol for division can be a slash, a line, or the division sign (<math>\div</math>), as in:

- <math>6/3\,</math> or <math>\frac 63</math> or <math>6 \div 3.</math>

Each, of those three, means "6 divided by 3" giving 2 as the answer. The first number is the **dividend** (6), and the second number is the **divisor** (3). The result (or answer) is the quotient. Whole numbers, any left-over amount is called the "remainder" (such as 14/4 gives 3 with the remainder as 2, as the number Template:Convert/numdisp, same as Template:Convert/numdisp or 3.5).

The numbers can be very big, such as with two hundred: 200/5 = 40, or with 7 billion: 7,000,000,000 / 1000 = 7,000,000 (equals 7 million).

## With multiplication

If *c* times *b* equals *a*, written as:

- <math>c \times b = a</math>

where *b* is not zero, then *a* divided by *b* equals *c*, written as:

- <math>\frac ab = c</math>

For instance,

- <math>\frac 63 = 2</math>

since

- <math>2 \times 3 = 6</math>.

In the above expression, *a* is called the **dividend**, *b* the **divisor** and *c* the **quotient**.

- <math>\frac x0 = ?</math>

...is not defined.

## Notation

Division is most often shown by placing the *dividend* over the *divisor* with a horizontal line, also called a vinculum, between them. For example, *a* divided by *b* is written

- <math>\frac ab.</math>

This can be read out loud as "a divided by b" or "a over b". A way to express division all on one line is to write the *dividend*, then a slash, then the *divisor*, like this:

- <math>a/b.\,</math>

This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters.

A typographical variation which is halfway between these two forms uses a slash but elevates the dividend, and lowers the divisor:

^{a}⁄_{b}.

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the *numerator* and *denominator*). A fraction is an accepted way of writing numbers. It is not always expected that the result of the division is written in decimals.

A less common way to show division is to use the obelus (or division sign) in this manner:

- <math>a \div b.</math>

But in elementary arithmetic this form is used rather often. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

In some non-English-speaking cultures, "*a* divided by *b*" is written *a* : *b*. However, in English usage the colon is restricted to expressing the related concept of ratios (with *a*:*b* said as "*a* is to *b*").

## Related pages

- Divisor, another meaning as a number which evenly divides an amount
- Division by two
- Remainder