Division (mathematics)

[math]\displaystyle{ 20/4=5 }[/math]

In mathematics, the word "division" means the operation which is the opposite of multiplication. The symbols for division are the slash ([math]\displaystyle{ / }[/math]) and the fraction line, as in:

[math]\displaystyle{ 6/3 }[/math] or [math]\displaystyle{ \frac 63 }[/math]

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, [math]\displaystyle{ 14/4 }[/math] gives quotient 3 with remainder 2, all expressed as the mixed number [math]\displaystyle{ 3\frac12 }[/math] or 3.5).

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

With multiplication

If [math]\displaystyle{ c }[/math] times [math]\displaystyle{ b }[/math] equals [math]\displaystyle{ a }[/math], written as:

[math]\displaystyle{ c \cdot b = a }[/math]

where [math]\displaystyle{ b }[/math] is not zero, then [math]\displaystyle{ a }[/math] divided by [math]\displaystyle{ b }[/math] equals [math]\displaystyle{ c }[/math], written as:

[math]\displaystyle{ \frac ab = c }[/math]

For instance,

[math]\displaystyle{ \frac 63 = 2 }[/math]

since

[math]\displaystyle{ 2 \cdot 3 = 6 }[/math].

In the above expression, [math]\displaystyle{ a }[/math] is called the dividend, [math]\displaystyle{ b }[/math] the divisor and [math]\displaystyle{ c }[/math] the quotient.

Division by zero, as in

[math]\displaystyle{ \frac x0 }[/math]

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, [math]\displaystyle{ a }[/math] divided by [math]\displaystyle{ b }[/math] is written as

[math]\displaystyle{ \frac ab }[/math]

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:

[math]\displaystyle{ a/b }[/math]

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 [math]\displaystyle{ a:b }[/math]. However, in English-speaking countries the colon is restricted to expressing the related concept of ratios (where [math]\displaystyle{ a:b }[/math] reads "a is to b").

Division (mathematics) Media

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  Plus and minuses. An obelus used as a variant of the minus sign in an excerpt from an official Norwegian trading statement form called «Næringsoppgave 1» for the taxation year 2010.

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