In mathematics, especially in elementary arithmetic, **division** is an arithmetic operation which is the inverse of multiplication. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
In mathematics, multiplication is an elementary arithmetic operation. ...
Specifically, if *c* times *b* equals *a*, written: where *b* is not zero, then *a* divided by *b* equals *c*, written: 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ...
For instance, since - .
In the above expression, *a* is called the **dividend**, *b* the **divisor** and *c* the **quotient**. Division by zero (i.e. where the divisor is zero) is not defined. In mathematics, a division is called a division by zero if the divisor is zero. ...
## 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 A vinculum is a horizontal line placed over a mathematical expression, used to indicate that it is to be considered a group. ...
This can be read out loud 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 slash or stroke, /, is a punctuation mark. ...
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 programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...
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 (although typically called the *numerator* and *denominator*), and there is no implication that the division needs to be evaluated further. For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ...
The integers are commonly denoted by the above symbol. ...
A less common way to show division is to use the obelus (or division sign) in this manner: A dagger (†, †, U+2020) is a typographical symbol or glyph. ...
This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator. A calculator is a device for performing calculations. ...
In some non-English-speaking cultures, "a divided by b" is written *a* : *b*. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b"). The English language is a West Germanic language that originates in England. ...
The colon (:) is a punctuation mark, visually consisting of two equally sized dots centered on the same vertical line. ...
A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ...
## 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. In mathematics, a multiplication table is used to define a multiplication operation for an algebraic system. ...
In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...
For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ...
Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus...
In modern computers, long division has been replaced by faster methods; see Division (digital). Several algorithms exist to perform division in digital designs. ...
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. For the finite element analysis software, see ABAQUS. An abacus (plurals abacuses or abaci), also called a counting frame, is a calculating tool for performing arithmetical processes, often constructed as a wooden frame with beads sliding on wires. ...
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. Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ...
The modular multiplicative inverse of a number n modulo p is a number m such that n-1 m (mod p). ...
## 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. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...
- Say that 26 cannot be divided by 10.
- Give the answer as a decimal fraction or a mixed number, so or . This is the approach usually taken in mathematics.
- Give the answer as an integer
*quotient* and a *remainder*, so remainder 6. - Give the integer quotient as the answer, so . 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. Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus...
In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ...
In mathematics, a quotient is the end result of a division problem. ...
In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder â€”an amount left overâ€” is also acknowledged. ...
A computer program is a collection of instructions that describe a task, or set of tasks, to be carried out by a computer. ...
A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...
C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ...
MATLAB is a numerical computing environment and programming language. ...
Names and symbols used for integer division include div, /, , and %. Definitions vary regarding integer division when the quotient is negative: rounding may be toward zero or toward minus infinity. The infinity symbol âˆž in several typefaces. ...
Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another. A divisibility rule is a method that can be used to determine whether a number is evenly divisible by other numbers. ...
## 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 In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
All four quantities are integers, and only *p* may be 0. This definition ensures that division is the inverse operation of multiplication. In mathematics, multiplication is an elementary arithmetic operation. ...
## 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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
## Division of complex numbers Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus: In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
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: 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. In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of lower degree, a generalized version of the familiar arithmetic technique called long division. ...
## Division of matrices One can define a division operation for matrices. The usual way to do this is to define *A* / *B* = *AB*^{−1}, where *B*^{−1} denotes the inverse of *B*. In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
### Left and right division Because matrix multiplication is not commutative, one can also define a left division or so-called *backslash-division* as *A* *B* = *A*^{−1}*B*. For this to be well defined, *B*^{−1} need not exist, however *A*^{−1} does need to exist. To avoid confusion, division as defined by *A* / *B* = *AB*^{−1} is sometimes called *right division* or *slash-division* in this context. This article gives an overview of the various ways to perform matrix multiplication. ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that division is always possible. ...
### Matrix division and pseudoinverse To avoid problems when *A*^{−1} and/or *B*^{−1} do not exist, division can also be defined as multiplication with the pseudoinverse, i.e., *A* / *B* = *AB*^{+} and *A* *B* = *A*^{+}*B*, where *A*^{+} and *B*^{+} denote the pseudoinverse of *A* and *B*. In mathematics, and in particular linear algebra, the pseudoinverse of an matrix is a generalization of the inverse matrix. ...
## Division in abstract algebra In abstract algebras such as matrix algebras and quaternion algebras, fractions such as are typically defined as or where *b* is presumed to be an invertible element (i.e. there exists a multiplicative inverse *b* ^{− 1} such that *b**b* ^{− 1} = *b* ^{− 1}*b* = 1 where 1 is the multiplicative identity). In an integral domain where such elements may not exist, *division* can still be performed on equations of the form *a**b* = *a**c* or *b**a* = *c**a* 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. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the 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**. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 â‰ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
The inspiration for the name of the principle: pigeons in holes. ...
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ...
In mathematics, the Bott periodicity theorem is a result from homotopy theory which was discovered by Raoul Bott during the latter part of the 1950s, and proved to be of foundational significance for much further research, in particular in K-theory. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || . || satisfying ||xy|| = ||x|| ||y|| for all x and y in A. While the definition allows normed division algebras to be infinite-dimensional, this, in...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In mathematics, the octonions are a nonassociative extension of the quaternions. ...
## Division and calculus The derivative of the quotient of two functions is given by the quotient rule: For a non-technical overview of the subject, see Calculus. ...
In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ...
There is no general method to integrate the quotient of two functions. The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically...
## See also In mathematics, an aliquot part (or simply aliquot) of an integer is any of its integer divisors. ...
Several algorithms exist to perform division in digital designs. ...
In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ...
The reciprocal function: y = 1/x. ...
In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...
Division by two is simple in even-numbered bases. ...
In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...
In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ...
This picture illustrates how the hours on a clock form a group under modular addition. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
A vinculum is a horizontal line placed over a mathematical expression, used to indicate that it is to be considered a group. ...
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