Division (mathematics)

In these enlarged number systems, division is the inverse operation to multiplication, that is a = c / b means a × b = c, as long as b is not zero.

Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate (which define multiplication and addition over single-variabled formulas).

Sometimes this remainder is added to the quotient as a fractional part, so 10 / 3 is equal to ⁠3+1/3⁠ or 3.33..., but in the context of integer division, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded or rounded).

Unlike multiplication and addition, division is not commutative, meaning that a / b is not always equal to b / a.

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.

A way to express division all on one line is to write the dividend (or numerator), then a slash, then the divisor (or denominator), as follows: This is the usual way of specifying division in most computer programming languages, since it can easily be typed as a simple sequence of ASCII characters.

Some mathematical software, such as MATLAB and GNU Octave, allows the operands to be written in the reverse order by using the backslash as the division operator: 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 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: This form is infrequent except in elementary arithmetic.

The obelus was introduced by Swiss mathematician Johann Rahn in 1659 in Teutsche Algebra.

[11] In some non-English-speaking countries, a colon is used to denote division:[12] This notation was introduced by Gottfried Wilhelm Leibniz in his 1684 Acta eruditorum.

However, in English usage the colon is restricted to expressing the related concept of ratios.

[13] 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' – a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself.

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 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).

The user is responsible, however, for mentally keeping track of the decimal point.

Such a case uses one of five approaches: Dividing integers in a computer program requires special care.

Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above.

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.

This definition ensures that division is the inverse operation of multiplication.

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

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 explicitly to avoid confusion.

Because matrix multiplication is not commutative, one can also define a left division or so-called backslash-division as A \ B = A−1B.

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

To avoid problems when A−1 and/or B−1 do not exist, division can also be defined as multiplication by the pseudoinverse.

In abstract algebra, given a magma with binary operation ∗ (which could nominally be termed multiplication), left division of b by a (written a \ b) is typically defined as the solution x to the equation a ∗ x = b, if this exists and is unique.

Division in this sense does not require ∗ to have any particular properties (such as commutativity, associativity, or an identity element).

A magma for which both a \ b and b / a exist and are unique for all a and all b (the Latin square property) is a quasigroup.

In a quasigroup, division in this sense is always possible, even without an identity element and hence without inverses.

However, in certain higher level mathematics division by zero is possible by the zero ring and algebras such as wheels.