# Dictionary Definition

divided adj

1 separated into parts or pieces; "opinions are
divided" [ant: united]

2 having a median strip or island between lanes
of traffic moving in opposite directions; "a divided highway" [syn:
dual-lane]

3 distributed in portions (often equal) on the
basis of a plan or purpose [syn: divided up,
shared, shared
out]

# User Contributed Dictionary

## English

### Pronunciation

### Verb

divided- past of divide

### Adjective

#### Translations

separated or split into pieces

- Finnish: jaettu

having conflicting interests or emotions

disunited

- Finnish: jakaantunut, hajanainen

separated into lanes, that move in opposite
directions, by a median

# Extensive Definition

In mathematics, especially in
elementary arithmetic, division is an
arithmetic operation which is the inverse of multiplication.

Specifically, if c times b equals a, written:

- c \times b = a\,

- \frac ab = c

- \frac 63 = 2

- 2 \times 3 = 6\,.

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.

## 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. For example, a divided by b is written

- \frac ab

- a/b\,

A typographical variation, which is 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 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.

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

- a \div b

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").

## Computing division

A person who knows the multiplication tables can divide two integers using pencil and paper and the method of long division. 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. 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).

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.

In modular
arithmetic, some numbers have a
multiplicative inverse with respect to the modulus. 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

The division algorithm is a theorem in mathematics which precisely expresses the outcome of the usual process of division of integers. 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.## 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; division becomes a partial function.
- Give the answer as a decimal fraction or a mixed number, so \frac = 2.6 or 26/10 = 2 \frac 35. This is the approach usually taken in mathematics.
- Give the answer as an integer quotient and a remainder, so \frac = 2 remainder 6.
- Give the integer quotient as the answer, so \frac = 2. This is sometimes called integer division.

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.

Divisibility
rules can sometimes be used to quickly determine whether one
integer divides exactly into another.

## 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

- = \times = .

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:

- = + i.

All four quantities are real numbers. 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 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 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.### 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. 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 are typically defined as a \cdot or a \cdot b^
where b is presumed to be an invertible element (i.e. there exists
a multiplicative inverse b^ such that bb^ = b^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 ab = ac or ba = ca 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.

## Division and calculus

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

- ' = \frac.

There is no general method to integrate the quotient of two
functions.

## See also

## External links

- Division on a Japanese abacus selected from Abacus: Mystery of the Bead
- Chinese Short Division Techniques on a Suan Pan
- Rules of divisibility

divided in Aymara: Jaljayaña

divided in Bulgarian: Деление

divided in Catalan: Divisió

divided in Czech: Dělení

divided in Welsh: Rhannu (mathemateg)

divided in Danish: Division (matematik)

divided in German: Division (Mathematik)

divided in Spanish: División (matemática)

divided in Esperanto: Divido

divided in Persian: تقسیم

divided in French: Division

divided in Scottish Gaelic: Roinn
(matamataig)

divided in Galician: División
(matemáticas)

divided in Korean: 나눗셈

divided in Icelandic: Deiling

divided in Italian: Divisione (matematica)

divided in Lithuanian: Dalyba

divided in Dutch: Delen

divided in Japanese: 除法

divided in Norwegian: Divisjon
(matematikk)

divided in Novial: Divisione

divided in Polish: Dzielenie

divided in Portuguese: Divisão

divided in Quechua: Rakiy

divided in Russian: Деление (математика)

divided in Sicilian: Spartuta

divided in Simple English: Division

divided in Slovak: Delenec

divided in Slovenian: Deljenje

divided in Serbian: Дељење

divided in Finnish: Jakolasku

divided in Swedish: Division (matematik)

divided in Tagalog: Dibisyon

divided in Tamil: வகுத்தல் (கணிதம்)

divided in Thai: การหาร

divided in Urdu: تقسیم (ریاضی)

divided in Chinese: 除法

# Synonyms, Antonyms and Related Words

alienated, bifurcated, bisected, branched, branching, cleft, cloven, detached, dichotomous, dimidiate, disaffected, disarticulated, disconnected, disengaged, disjoined, disjoint, disjointed, disjunct, dislocated, dispersed, disunited, divorced, estranged, forked, forking, halved, irreconcilable, isolated, ramified, removed, riven, scattered, segregated, separated, sequestered, shut off,
split, torn