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Division is one of the four basic operations of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, the ways that numbers are combined to make new numbers. The other operations are
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
, subtraction, and
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
. At an elementary level the division of two
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s is, among other possible interpretations, the process of calculating the number of times one number is contained within another. This number of times need not be an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). The division with remainder or Euclidean division of two
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
provides an integer ''quotient'', which is the number of times the second number is completely contained in the first number, and a ''remainder'', which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. For example, if 21 apples are divided between 4 people, everyone receives 5 apples again, and 1 apple remains. For division to always yield one number rather than a quotient plus a remainder, the natural numbers must be extended to
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s or
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. In these enlarged number systems, division is the inverse operation to multiplication, that is means , as long as is not zero. If , then this is a division by zero, which is not defined. In the 21-apples example, everyone would receive 5 apple and a quarter of an apple, thus avoiding any leftover. Both forms of division appear in various
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
s, different ways of defining mathematical structure. Those in which a Euclidean division (with remainder) is defined are called
Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers ...
s and include
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
s in one indeterminate (which define multiplication and addition over single-variabled formulas). Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings. In a ring the elements by which division is always possible are called the units (for example, 1 and −1 in the ring of integers). Another generalization of division to algebraic structures is the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
, in which the result of "division" is a group rather than a number.


Introduction

The simplest way of viewing division is in terms of
quotition and partition In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotition division one asks, "how many parts are there?"; While in partition division one asks, "what is the size of each part?". For example, the expressio ...
: from the quotition perspective, means the number of 5s that must be added to get 20. In terms of partition, means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that ''twenty divided by five is equal to four''. This is denoted as , or . What is being divided is called the ''dividend'', which is divided by the ''divisor'', and the result is called the ''quotient''. In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient. Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example, leaves a remainder of 1, as 10 is not a multiple of 3. Sometimes this remainder is added to the quotient as a
fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part ca ...
, so is equal to or , but in the context of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
division, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded or rounded). When the remainder is kept as a fraction, it leads to a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
. The set of all rational numbers is created by extending the integers with all possible results of divisions of integers. Unlike multiplication and addition, division is not commutative, meaning that is not always equal to . Division is also not, in general, associative, meaning that when dividing multiple times, the order of division can change the result. For example, , but (where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses). Division is traditionally considered as left-associative. That is, if there are multiple divisions in a row, the order of calculation goes from left to right:George Mark Bergman
Order of arithmetic operations
Education Place

: a / b / c = (a / b) / c = a / (b \times c) \;\ne\; a/(b/c)= (a\times c)/b. Division is right-distributive over addition and subtraction, in the sense that : \frac = (a \pm b) / c = (a/c)\pm (b/c) =\frac \pm \frac. This is the same for
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
, as (a + b) \times c = a \times c + b \times c. However, division is ''not'' left-distributive, as : \frac = a / (b + c) \;\ne\; (a/b) + (a/c) = \frac.   For example \frac = \frac = 2 , but \frac + \frac = 6+3 = 9 . This is unlike the case in multiplication, which is both left-distributive and right-distributive, and thus distributive.


Notation

Division is often shown in algebra and science by placing the ''dividend'' over the ''divisor'' with a horizontal line, also called a fraction bar, between them. For example, "''a'' divided by ''b''" can written as: :\frac ab which can also be read out loud as "divide ''a'' 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 Slash may refer to: * Slash (punctuation), the "/" character Arts and entertainment Fictional characters * Slash (Marvel Comics) * Slash (''Teenage Mutant Ninja Turtles'') Music * Harry Slash & The Slashtones, an American rock band * Nash ...
, then the ''divisor'' (or denominator), as follows: :a/b This is the usual way of specifying division in most computer
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
s, since it can easily be typed as a simple sequence of
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
characters. (It is also the only notation used for quotient objects in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
.) Some mathematical software, such as
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
and
GNU Octave GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a lan ...
, allows the operands to be written in the reverse order by using the backslash as the division operator: :b\backslash a A typographical variation halfway between these two forms uses a solidus (fraction slash), but elevates the dividend and lowers the divisor: :^\!/_ Any of these forms can be used to display a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. A fraction is a division expression where both dividend and divisor are
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s (typically called the ''numerator'' and ''denominator''), and there is no implication that the division must be evaluated further. A second way to show division is to use the division sign (÷, also known as obelus though the term has additional meanings), common in arithmetic, in this manner: :a \div b This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. This division sign is also used alone to represent the division operation itself, as for instance as a label on a key of a
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
. The obelus was introduced by Swiss mathematician
Johann Rahn Johann Rahn (Latinised form Rhonius) (10 March 1622 – 25 May 1676) was a Swiss mathematician who is credited with the first use of the division sign, ÷ (a repurposed obelus variant) and the therefore sign, ∴. The symbols were used in '' ...
in 1659 in ''Teutsche Algebra''. The ÷ symbol is used to indicate subtraction in some European countries, so its use may be misunderstood. In some non-
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
-speaking countries, a colon is used to denote division: :a : b This notation was introduced by
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
in his 1684 ''Acta eruditorum''. Leibniz disliked having separate symbols for ratio and division. However, in English usage the colon is restricted to expressing the related concept of
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s. Since the 19th century, US textbooks have used b)a or b \overline to denote ''a'' divided by ''b'', especially when discussing long division. The history of this notation is not entirely clear because it evolved over time.


Computing


Manual methods

Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of lollies, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of '
chunking Chunking may mean: * Chunking (division), an approach for doing simple mathematical division sums, by repeated subtraction * Chunking (computational linguistics), a method for parsing natural language sentences into partial syntactic structures * ...
' a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself. By allowing one to subtract more multiples than what the partial remainder allows at a given stage, more flexible methods, such as the bidirectional variant of chunking, can be developed as well. More systematically and more efficiently, two integers can be divided with pencil and paper with the method of short division, if the divisor is small, or long division, if the divisor is larger. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the procedure 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, which can make the problem easier to solve (e.g., 10/2.5 = 100/25 = 4). Division can be calculated with an abacus. Logarithm tables can be used to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result. Division can be calculated with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.


By computer

Modern
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s and
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...
s compute division either by methods similar to long division, or by faster methods; see Division algorithm. In
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
(modulo a prime number) and for
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
, nonzero numbers have a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
. In these cases, a division by may be computed as the product by the multiplicative inverse of . This approach is often associated with the faster methods in computer arithmetic.


Division in different contexts


Euclidean division

Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, ''a'', the ''dividend'', and ''b'', the ''divisor'', such that ''b'' ≠ 0, there are unique integers ''q'', the ''quotient'', and ''r'', the remainder, such that ''a'' = ''bq'' + ''r'' and 0 ≤ ''r'' < , where denotes the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of ''b''.


Of integers

Integers are not
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
under division. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches: # Say that 26 cannot be divided by 11; division becomes a partial function. # Give an approximate answer as a floating-point number. This is the approach usually taken in numerical computation. # Give the answer as a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
representing a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
, so the result of the division of 26 by 11 is \tfrac (or as a mixed number, so \tfrac = 2 \tfrac 4.) Usually the resulting fraction should be simplified: the result of the division of 52 by 22 is also \tfrac. This simplification may be done by factoring out the greatest common divisor. # Give the answer as an integer '' quotient'' and a '' remainder'', so \tfrac = 2 \mbox 4. To make the distinction with the previous case, this division, with two integers as result, is sometimes called '' Euclidean division'', because it is the basis of the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an e ...
. # Give the integer quotient as the answer, so \tfrac = 2. This is the '' floor function'' applied to case 2 or 3. It is sometimes called integer division, and denoted by "//". Dividing integers in a
computer program A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. Computer programs are one component of software, which also includes software documentation, documentation and oth ...
requires special care. Some
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
s treat integer division as in case 5 above, so the answer is an integer. Other languages, such as
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
and every
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...
return a rational number as the answer, as in case 3 above. These languages also provide functions to get the results of the other cases, either directly or from the result of case 3. Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative:
rounding Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $ with $, the fraction 312/937 with 1/3, or the expression with . Rounding is often done to ob ...
may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see
modulo operation In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
for the details. Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.


Of rational numbers

The result of dividing two
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s is another rational number when the divisor is not 0. The division of two rational numbers ''p''/''q'' and ''r''/''s'' can be computed as : = \times = . All four quantities are integers, and only ''p'' may be 0. This definition ensures that division is the inverse operation of
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
.


Of real numbers

Division of two
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s results in another real number (when the divisor is nonzero). It is defined such that ''a''/''b'' = ''c'' if and only if ''a'' = ''cb'' and ''b'' ≠ 0.


Of complex numbers

Dividing two
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (when the divisor is nonzero) results in another complex number, which is found using the conjugate of the denominator: : = = = + i. This process of multiplying and dividing by r-is is called 'realisation' or (by analogy) rationalisation. All four quantities ''p'', ''q'', ''r'', ''s'' are real numbers, and ''r'' and ''s'' may not both be 0. Division for complex numbers expressed in polar form is simpler than the definition above: : = = e^. Again all four quantities ''p'', ''q'', ''r'', ''s'' are real numbers, and ''r'' may not be 0.


Of polynomials

One can define the division operation for
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s in one variable over a field. Then, as in the case of integers, one has a remainder. See
Euclidean division of polynomials In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common ...
, and, for hand-written computation,
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 generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, bec ...
or synthetic division.


Of matrices

One can define a division operation for matrices. The usual way to do this is to define , where denotes the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
of ''B'', but it is far more common to write out explicitly to avoid confusion. An elementwise division can also be defined in terms of the Hadamard product.


Left and right division

Because matrix multiplication is not commutative, one can also define a
left division Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication. At an elementary level the division of two natural numbe ...
or so-called ''backslash-division'' as . For this to be well defined, need not exist, however does need to exist. To avoid confusion, division as defined by is sometimes called ''right division'' or ''slash-division'' in this context. Note that with left and right division defined this way, is in general not the same as , nor is the same as . However, it holds that and .


Pseudoinverse

To avoid problems when and/or do not exist, division can also be defined as multiplication by the pseudoinverse. That is, and , where and denote the pseudoinverses of ''A'' and ''B''.


Abstract algebra

In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, given a
magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...
with binary operation ∗ (which could nominally be termed multiplication), left division of ''b'' by ''a'' (written ) is typically defined as the solution ''x'' to the equation , if this exists and is unique. Similarly, right division of ''b'' by ''a'' (written ) is the solution ''y'' to the equation . Division in this sense does not require ∗ to have any particular properties (such as commutativity, associativity, or an identity element). "Division" in the sense of "cancellation" can be done in any magma by an element with the cancellation property. Examples include
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
algebras and
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
algebras. A quasigroup is a structure in which division is always possible, even without an identity element and hence inverses. In an integral domain, where not every element need have an inverse, ''division'' by a cancellative element ''a'' can still be performed on elements of the form ''ab'' or ''ca'' by left or right cancellation, respectively. If a ring is finite and every nonzero element is cancellative, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, and ''division'' by any nonzero element is possible. 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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to either the real numbers R, the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s C, the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s H, or the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s O.


Calculus

The
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the quotient of two functions is given by the quotient rule: :' = \frac.


Division by zero

Division of any number by
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
in most mathematical systems is undefined, because zero multiplied by any finite number always results in a product of zero. Entry of such an expression into most
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s produces an error message. However, in certain higher level mathematics division by zero is possible by the zero ring and algebras such as
wheels A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to b ...
.Jesper Carlström
"On Division by Zero"
Retrieved October 23, 2018
In these algebras, the meaning of division is different from traditional definitions.


See also

* 400AD Sunzi division algorithm * Division by two * Galley division * Inverse element *
Order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For examp ...
*
Repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational i ...


Notes


References


External links


Planetmath division


selected fro
Abacus: Mystery of the Bead

Chinese Short Division Techniques on a Suan Pan


{{Authority control Elementary arithmetic