Division (mathematics)

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:

^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 (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").

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