# 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

In mathematics, the word "division" means the operation which is the opposite of multiplication. The symbols for division are the slash ($/$ ) and the fraction line, as in: $20/4=5$ $6/3$ or ${\frac {6}{3}}$ where each of the three expressions means "6 divided by 3", with 2 as the answer. The first number is the dividend (6), and the second number is the divisor (3). The result (or answer) of a division is the quotient, where any left-over amount as whole numbers is called the "remainder". For example, $14/4$ gives quotient 3 with remainder 2, all expressed as the mixed number $3{\frac {1}{2}}$ or 3.5).

The numbers involved in division can be very big, such as the case with two hundred: $200/5=40$ , or with 7 billion: $7,000,000,000/1000=7,000,000$ (where the quotient is equal to 7 million).

## With multiplication

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

$c\cdot b=a$

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

$2\cdot 3=6$ .

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

Division by zero, as in

${\frac {x}{0}}$

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 as

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

This can be read 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 (in which case, the two numbers are typically referred to as 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.

In some non-English-speaking cultures, "a divided by b" is written as $a:b$ . However, in English-speaking countries the colon is restricted to expressing the related concept of ratios (where $a:b$  reads "a is to b").