Division

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This article is about the arithmetic operation. For other uses, see Division (disambiguation).

In mathematics, especially elementary arithmetic, division is an arithmetic operation which is the reverse operation of multiplication and sometimes can be interpreted as repeated subtraction.

Specifically, if

a × b = c,

where b is nonzero, then

a = c ÷ b

(read as "c divided by b"). So for instance, 6 ÷ 3 = 2 since 2 × 3 = 6.

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

The expression c ÷ b is also written "c/b" (read "c over b"), especially in higher mathematics (including applications to science and engineering) and in computer programming languages. This form is also often used as the final form of a fraction, without any implication that it needs to be evaluated further.

In most non-English languages, c ÷ b is written c : b. In English usage the colon is restricted to the related concept of ratios.

The meaning of division by zero is not usually defined.

Computing division

With a knowledge of multiplication tables, two integers can be divided on paper using the method of long division. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.

Division can be calculated with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.

In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. In such a case, division can be calculated by multiplication. This approach is useful in computers that do not have a fast division instruction.

Division of integers

Division of integers is not closed; apart from division by zero being undefined, the quotient will not be an integer unless the dividend is an integer multiple of the divisor; for example 26 cannot be divided by 10 to give an integer. In such a case there are three possible approaches.

1. Say that 26 cannot be divided by 10.
2. Give the answer as a decimal fraction or a mixed number, so 26 ÷ 10 = 2.6 or <math>26 / 10 = 2\frac{3}{5}<math>. This is the approach usually taken in mathematics.
3. Give the answer as a quotient and a remainder, so 26 ÷ 10 = 2 remainder 6. This approach is often used in computer science. In some computer integer arithmetic, 26/10 (or 26i / 10i) is given as 2 while 26 modulo 10 (or 26i % 10i) is given as 6.

Division of rational numbers

The result of dividing two rational numbers is another rational number when the divisor is not 0. We may define division of two rational numbers p/q and r/s by

<math>{p/q \over r/s} = (p \times s)/(q \times r)<math>

All four quantities are integers, and only p may be 0. This definition ensures that division is the inverse operation of multiplication.

Division of real numbers

Division of two real numbers results in another real number when the divisor is not 0. It is defined such a/b = c if and only if a = cb and b ≠ 0.

Division of complex numbers

Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus:

<math>{p + iq \over r + is} = {pr + qs \over r^2 + s^2} + i{qr - ps \over r^2 + s^2}<math>

All four quantities are real numbers. r and s may not both be 0.

Division for complex numbers expressed in polar form is simpler and easier to remember than the definition above:

<math>{pe^{iq} \over re^{is}} = {p \over r}e^{i(q - s)}<math>

Again all four quantities are real numbers. r may not be 0.

Division in abstract algebra

In abstract algebras such as matrix algebras and quaternion algebras, fractions such as <math>{a \over b}<math> are typically defined as <math>a \cdot {1 \over b}<math> or <math>a \cdot b^{-1}<math> where <math>b<math> is presumed to be an invertible element (i.e. there exists a multiplicative inverse <math>b^{-1}<math> such that <math>bb^{-1} = b^{-1}b = 1<math> where <math>1<math> is the multiplicative identity). In an integral domain where such elements may not exist, division can still be performed on equations of the form <math>ab = ac<math> or <math>ba = ca<math> by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. By a theorem of Wedderburn, all finite division rings are fields, hence every nonzero element of such a ring is invertible, so division by any nonzero element is possible in such a ring. To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.

Division and calculus

The derivative of the quotient of two functions is given by the quotient rule:

<math>{d \over dx} {f(x) \over g(x)} = {{d \over dx} f(x) \cdot g(x) - f(x) \cdot {d \over dx} g(x) \over (g(x))^2}<math>

There is no general method to integrate the quotient of two functions.

External links

Printable Worksheets for Practicing Division

See also: Rational number, Vulgar fraction, Reciprocal, Inverse element, Division by two, Division by zero, Quasigroup, Group, Field (algebra), Division algebra, Division ring, Long division, Vinculum

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