Division (mathematics)

$20:4=5$

In mathematics, the word "division" means the operation which is the opposite of multiplication. Some symbols for division can be a slash ($/$), a (fraction) line, or the division sign ($\div$), as in:

$6/3\,$    or    ${\frac {6}{3}}$    or    $6\div 3.$

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) 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 number 3+²⁄₄, or as 3+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:

$c\times b=a$

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.

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.

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 in the case of a label on a key of a calculator.

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