# Division (mathematics)

divisiondividingdividedinteger divisiondividedividendDIVdivided bydividesdivisible
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers.wikipedia
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The other operations are addition, subtraction, and multiplication (which can be viewed as the inverse of division).
Addition (usually signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

### Arithmetic

arithmetic operationsarithmeticsarithmetic operation
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers.
Arithmetic (from the Greek ἀριθμός arithmos, "number" and τική [τέχνη], tiké [téchne], "art") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

### Obelus

obeli÷Division sign
Several symbols are used for the division operator, including the obelus, the colon and the slash .
In mathematics it is mainly used to represent the mathematical operation of division.

### Multiplication

productmultipliermultiplying
The other operations are addition, subtraction, and multiplication (which can be viewed as the inverse of division).
Multiplication (often denoted by the cross symbol "×", by the dot "⋅", by juxtaposition, or, on computers, by an asterisk "*") is one of the four elementary mathematical operations of arithmetic, with the others being addition, subtraction and division.

### Euclidean division

Division theoremdividedivisible by two without remainder
The division with remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, 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.
In arithmetic, Euclidean division — or division with remainder — is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor.

### Division by zero

divide by zerodividing by zerodivided by zero
, then this is a division by zero, which is not defined.
In mathematics, division by zero is division where the divisor (denominator) is zero.

### Number

number systemnumericalnumbers
In these enlarged number systems, division is the inverse operation to multiplication, that is
Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation.

### Field (mathematics)

fieldfieldsfield theory
Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings.
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.

### Division ring

skew fieldskew-fieldskewfield
Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings.
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

### Remainder

in remainderlaver
Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example,
In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division).

### Integer

integersintegralZ
This number of times is not always an integer (a number that can be obtained using the other arithmetic operations on the natural numbers), which led to two different concepts.
is not closed under division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.

### Commutative property

commutativecommutativitycommute
Unlike multiplication and addition, Division is not commutative, meaning that
The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations.

### Quotition and partition

possible interpretations
At an elementary level the division of two natural numbers is – among other possible interpretations – the process of calculating the number of times one number is contained within another one.
Division involves thinking about a whole in terms of its parts.

### Long division

(long)Division tableauSchoolbook long division
Since the 19th century US textbooks have used b)~a or to denote a divided by b, especially when discussing long division. More systematic and more efficient (but also more formalised, more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers with pencil and paper using the method of short division, if the divisor is small, or long division, if the divisor is larger.
It breaks down a division problem into a series of easier steps.

### Slide rule

slide rulescircular slide ruleslide-rule
A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale.
The slide rule is used primarily for multiplication and division, and also for functions such as exponents, roots, logarithms, and trigonometry, but typically not for addition or subtraction.

### Calculator

pocket calculatorcalculatorselectronic calculator
The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

### Slash (punctuation)

slash/solidus
Several symbols are used for the division operator, including the obelus, the colon and the slash .
The division slash, is used between two numbers to indicate division (e.g., 23÷43 can also be written as 23 ∕ 43).

### Short division

More systematic and more efficient (but also more formalised, more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers with pencil and paper using the method of short division, if the divisor is small, or long division, if the divisor is larger.
In arithmetic, short division is a division algorithm which breaks down a division problem into a series of easy steps.

### Modulo operation

modulomodmodulus
Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see Modulo operation for the details.
In computing, the modulo operation finds the remainder after division of one number by another (called the modulus of the operation).

### Chunking (division)

Chunking
Distributing the objects several at a time in each round of sharing to each portion leads to the idea of "chunking" — a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself.
In mathematics education at the primary school level, chunking (sometimes also called the partial quotients method) is an elementary approach for solving simple division questions by repeated subtraction.

### Natural number

natural numberspositive integerpositive integers
The division with remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, 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. At an elementary level the division of two natural numbers is – among other possible interpretations – the process of calculating the number of times one number is contained within another one.
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers

### Fraction (mathematics)

denominatorfractionsfraction
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.
Other uses for fractions are to represent ratios and division.

### Greatest common divisor

gcdcommon factorgreatest common factor
In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

### Quotient

In arithmetic, a quotient (from "how many times", pronounced ) is the quantity produced by the division of two numbers.

### Euclidean algorithm

Euclid's algorithmEuclideanEuclid
It is used for reducing fractions to their simplest form and for performing division in modular arithmetic.