Division (mathematics)

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone

Specifically, if c times b equals a, written:

$c \times b = a\,$

where b is not zero, then a divided by b equals c, written:

$\frac ab = c$

For instance,

$\frac 63 = 2$

since

$2 \times 3 = 6\,$ .

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

Division by zero (i. In e. where the divisor is zero) is not defined.

1 Notation
2 Computing division
3 Division algorithm
4 Division of integers
5 Division of rational numbers
6 Division of real numbers
7 Division of complex numbers
8 Division of polynomials
9 Division of matrices
- 9.1 Left and right division
- 9.2 Matrix division and pseudoinverse
10 Division in abstract algebra
11 Division and calculus
12 See also
13 External links

Notation

Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a vinculum or fraction bar, between them. A vinculum is a horizontal line placed over a Mathematical expression, used to indicate that it is to be considered a group For example, a divided by b is written

$\frac ab$

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, or numerator then a slash, then the divisor, or denominator 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. The slash ( /) is a punctuation mark It is also called a virgule, diagonal, stroke, forward slash, oblique dash, A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer.

A typographical variation, which is halfway between these two forms, uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:

^a⁄_b

Any of these forms can be used to display a fraction. The solidus ( ⁄) is a punctuation mark that is not found on standard keyboards In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object 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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:

$a \div b$

This form is infrequent except in elementary arithmetic. The word " obelus " is also an alternative name for the dagger († symbol 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 device for performing mathematical calculations distinguished from a Computer by having a limited problem solving ability and an interface optimized for interactive

In some non-English-speaking cultures, "a divided by b" is written a : b. English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b"). A ratio is an expression which compares quantities relative to each other

Computing division

A person who knows the multiplication tables can divide two integers using pencil and paper and the method of long division. Long Division is the second album by the Rustic Overtones, originally released on November 17 1995 If the dividend has a fractional part (expressed as a decimal fraction), we can continue the algorithm past the ones place as far as desired. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object The decimal ( base ten or occasionally denary) Numeral system has ten as its base. If the divisor has a fractional part, we can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.

Modern computers compute division by methods that are faster than long division: see Division (digital). Several algorithms exist to perform division in digital designs

A person can calculate division 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. An abacus, also called a counting frame, is a calculating tool used primarily by Asians for performing arithmetic processes

In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers The modular multiplicative inverse of an Integer n modulo p is an integer m such that n -1 ≡ We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.

Division algorithm

Main article: Division algorithm

The division algorithm is a theorem in mathematics which precisely expresses the outcome of the usual process of division of integers. The division algorithm is a Theorem in Mathematics which precisely expresses the outcome of the usual process of division of Integers The name The division algorithm is a Theorem in Mathematics which precisely expresses the outcome of the usual process of division of Integers The name In particular, the theorem asserts that integers called the quotient q and remainder r always exist and that they are uniquely determined by the dividend a and divisor d, with d ≠ 0. Formally, the theorem is stated as follows: There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d. In Mathematics and Logic, the phrase "there is one and only one " is used to indicate that exactly one object with a certain property exists In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

Division of integers

Division of integers is not closed. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set 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; division becomes a partial function. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function
Give the answer as a decimal fraction or a mixed number, so $\frac{26}{10} = 2.6$ or $26/10 = 2 \frac 35$ . The decimal ( base ten or occasionally denary) Numeral system has ten as its base. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object This is the approach usually taken in mathematics.
Give the answer as an integer quotient and a remainder, so $\frac{26}{10} = 2$ remainder 6. In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left
Give the integer quotient as the answer, so $\frac{26}{10} = 2$ . This is sometimes called integer division.

One has to be careful when performing division of integers in a computer program. Computer programs (also software programs, or just programs) are instructions for a Computer. Some programming languages, such as C, will treat division of integers as in case 4 above, so the answer will be an integer. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured 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. 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. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness

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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions We may define division of two rational numbers p/q and r/s by

${p/q \over r/s} = {p \over q} \times {s \over r} = {ps \over qr}.$

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. In Mathematics, the real numbers may be described informally in several different ways 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:

${p + iq \over r + is} = {p + qs \over r^2 + s^2} + i{qr - ps \over r^2 + s^2}.$

All four quantities are real numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted r and s may not both be 0.

Division for complex numbers expressed in polar form is simpler than the definition above:

${pe^{iq} \over re^{is}} = {p \over r}e^{i(q - s)}.$

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

Division of polynomials

One can define the division operation for polynomials. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Then, as in the case of integers, one has a remainder. See polynomial long division. In Algebra, polynomial long division is an Algorithm for dividing a Polynomial by another polynomial of the same or lower degree, a generalised

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, but it is far more common to write out AB⁻¹ (or B⁻¹A) explicitly to avoid confusion. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by-

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⁻¹B. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division 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.

Note that with left and right division defined this way, A/(BC) is in general not the same as (A/B)/C and nor is (AB)\C the same as A\(B\C), but A/(BC) = (A/C)/B and (AB)\C = B\(A\C).

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.

Division in abstract algebra

In abstract algebras such as matrix algebras and quaternion algebras, fractions such as ${a \over b}$ are typically defined as $a \cdot {1 \over b}$ or $a \cdot b^{-1}$ where $b$ is presumed to be an invertible element (i. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician 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. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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. The pigeonhole principle, also known as Dirichlet's box (or drawer) principle, states that given two Natural numbers n and To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. 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 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. In Mathematics, the Bott periodicity theorem is a result from Homotopy theory discovered by Raoul Bott during the latter part of the 1950s which proved In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a normed division algebra A is a Division algebra over the real or complex numbers which is also a Normed vector In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real

Division and calculus

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

${\left(\frac fg\right)}' = \frac{f'g - fg'}{g^2}.$

There is no general method to integrate the quotient of two functions. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change 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 The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

External links

In Mathematics, an aliquot part (or simply aliquot) of an integer is any of its Integer Proper divisors For instance 2 is an Several algorithms exist to perform division in digital designs In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to Division by Two is simple in even-numbered bases NOTE The following methods return only the Integer part of the result Long Division is the second album by the Rustic Overtones, originally released on November 17 1995 In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division A vinculum is a horizontal line placed over a Mathematical expression, used to indicate that it is to be considered a group A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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