Division (mathematics)

In mathematics, division is 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 both expressions mean "6 divided by 3", with 2 as the answer. The first number (6) is the dividend, and the second number (3) is the divisor. 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 $3.5$).

With multiplication

If $x$ times $y$ equals $z$, written as:

$x\cdot y=z$

where $y$ is not $0$, then $x$ equals $z$ divided by $y$, written as:

$x={\frac {z}{y}}$

For example,

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

because

$2\cdot 3=6$.

In the above expressions, the dividend is $6$, the divisor is $3$ and the quotient is $2$ (with no remainder).

Division by zero, as in

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

is undefined (possibly infinity).

Also if you wrote:

$0/0$

it is still undefined (because if you divide by 0 is undefined).

Notation

Division is most often shown by placing the dividend over the divisor with a horizontal line (called a vinculum) between them. For example, $x$ divided by $y$ is written as

${\frac {x}{y}}$

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

$x/y$

This is the usual way to write 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 show a fraction. A fraction is a division expression where the dividend and divisor are both integers (in that case, the two numbers are called numerator and denominator).

Division should not be confused with ratios, which are written as $a:b$, and read "$a$ (is) to $b$".

Related pages

• Divisor, another meaning as a number which evenly divides an amount
• Division by two
• Long division
• Modular arithmetic
• Remainder

Other websites

• Division on a Japanese abacus Archived 2009-04-16 at the Wayback Machine selected from Abacus: Mystery of the Bead
• Chinese Short Division Techniques on a Suan Pan Archived 2015-05-03 at the Wayback Machine
• Rules of divisibility Archived 2015-05-03 at the Wayback Machine