Division (mathematics)
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- This article is about the arithmetic operation. For other uses, see Division (disambiguation).
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the reverse operation of multiplication, and sometimes it can be interpreted as repeated subtraction.
Specifically, if
- a × b = c,
where b is not zero, then
- a = c ÷ b
(read as "c divided by b"). 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.
Division by zero is usually not defined.
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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 four possible approaches.
- Say that 26 cannot be divided by 10.
- 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.
- Give the answer as a quotient and a remainder, so 26 ÷ 10 = 2 remainder 6.
- Give the quotient as the answer, so 26 ÷ 10 = 2. This is sometimes called integer division.
One has to be careful when performing division of integers in a computer program. Some programming languages, such as C, will treat division of integers as in case 4 above, so the answer will be an integer. Other languages, such as MATLAB, will first convert the integers to real numbers, and then give a real number as the answer, as in case 2 above.
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 of polynomials
One can define the division operation for polynomials. Then, as in the case of integers, one has a remainder. See polynomial long division.
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>\left({f \over g}\right)' = {f'g - fg' \over g^2}.<math>
There is no general method to integrate the quotient of two functions.
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
External links
- Template:Planetmath reference
- Printable Worksheets for Practicing Division (http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1317&Method=Worksheet&NQ=24&NQ4P=3)
- Division on a Japanese abacus (http://webhome.idirect.com/~totton/abacus/pages.htm#Division1) selected from Abacus: Mystery of the Bead (http://webhome.idirect.com/~totton/abacus/)
- Chinese Short Division Techniques on a Suan Pan (http://webhome.idirect.com/~totton/suanpan/sh_div/)
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