# Division (mathematics)

arithmetic operation; one of the four basic operations of arithmetic (others being addition, subtraction, multiplication).The division of two natural numbers is the process of calculating the number of times one number is contained within one another $20:4=5$ 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 ($\div$ ), as in:

$6/3\,$ or    ${\frac {6}{3}}$ or    $6\div 3.$ 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 324, same as 312 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:

$c\times b=a$

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

${\frac {a}{b}}=c$

For instance,

${\frac {6}{3}}=2$

since

$2\times 3=6$ .

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

${\frac {x}{0}}=?$

...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

${\frac {a}{b}}.$

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:

$a/b.\,$

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:

ab .

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:

$a\div b.$

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").