# division

## Division

[https://en.wikipedia.org/wiki/Division\_(mathematics](https://en.wikipedia.org/wiki/Division_\(mathematics))

* division is one of the 4 basic operations of arithmetic
* division of dividend n by a divisor d is denoted by n/d, or n ÷ d
* division is undefined if the divisor d is 0
* multiplication and division are each other's inverse operations
* multiplication subsumes division, n ÷ d = n _1/d = n_ d⁻¹
* division escapes the closure but multiplication maintains within it
* kinds of division:
  * precise: Euclidean division (closed) and proper division (open)
  * imprecise: integer division (Euclidean sans remainder)

Division of two integers, n ÷ d, may be considered as the process of calculating the number of times d is contained within n. Since the result is not always an integer (divison escapes the closure), it has spawned two different concepts of division: _proper division_ whose codomain is ℝ, and _integer division_ that drops the fractional part from the result in order to force codomain to ℤ (to preserve the closure over ℤ).

∀n,d ∈ ℤ -> n/d ∈ ℝ 7 / 4 = 1.75\
∀n,d ∈ ℤ -> n//d ∈ ℤ 7 // 4 = 1

The _division with remainder_, or _Euclidean division_, of two natural numbers produces a _quotient_ and a _remainder_ such that `n = dq + r`. When the remainder is 0, the dividend n is divisible by the divisor d, `d|n`.

Proper division produces a result that is ℚ or ℝ, so all involved numbers must be in this domain as well. Also, divisor d must not be 0, in which case division is undefined, thus making division a partial function. In these enlarged number sets, division is the inverse operation to multiplication.

Both forms of division appear in various algebraic structures:

Those algebraic structures in which a Euclidean division is defined are called _Euclidean domains_, and include polynomial rings 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_, in which the result of "division" is a group rather than a number.

## Refs

[https://en.wikipedia.org/wiki/Division\_(mathematics](https://en.wikipedia.org/wiki/Division_\(mathematics)) [https://en.wikipedia.org/wiki/Divisor](https://en.wikipedia.org/wiki/Divisor) [https://en.wikipedia.org/wiki/Quotient](https://en.wikipedia.org/wiki/Quotient) [https://en.wikipedia.org/wiki/Remainder](https://en.wikipedia.org/wiki/Remainder) [https://en.wikipedia.org/wiki/Quotition\_and\_partition](https://en.wikipedia.org/wiki/Quotition_and_partition) [https://en.wikipedia.org/wiki/Aliquot\_sequence](https://en.wikipedia.org/wiki/Aliquot_sequence) [https://en.wikipedia.org/wiki/Aliquot\_sum](https://en.wikipedia.org/wiki/Aliquot_sum) [https://en.wikipedia.org/wiki/Division\_sign](https://en.wikipedia.org/wiki/Division_sign) [https://en.wikipedia.org/wiki/Euclidean\_division](https://en.wikipedia.org/wiki/Euclidean_division) [https://en.wikipedia.org/wiki/Division\_algorithm](https://en.wikipedia.org/wiki/Division_algorithm) [https://en.wikipedia.org/wiki/Greatest\_common\_divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) [https://en.wikipedia.org/wiki/Euclidean\_algorithm](https://en.wikipedia.org/wiki/Euclidean_algorithm) [https://en.wikipedia.org/wiki/Long\_division](https://en.wikipedia.org/wiki/Long_division) [https://en.wikipedia.org/wiki/Modular\_arithmetic](https://en.wikipedia.org/wiki/Modular_arithmetic) [https://en.wikipedia.org/wiki/Modulo\_operation](https://en.wikipedia.org/wiki/Modulo_operation) [https://en.wikipedia.org/wiki/Rod\_calculus](https://en.wikipedia.org/wiki/Rod_calculus) [https://en.wikipedia.org/wiki/Division\_by\_two](https://en.wikipedia.org/wiki/Division_by_two) [https://en.wikipedia.org/wiki/Galley\_division](https://en.wikipedia.org/wiki/Galley_division) [https://en.wikipedia.org/wiki/Inverse\_element](https://en.wikipedia.org/wiki/Inverse_element) [https://en.wikipedia.org/wiki/Order\_of\_operations](https://en.wikipedia.org/wiki/Order_of_operations) [https://en.wikipedia.org/wiki/Repeating\_decimal](https://en.wikipedia.org/wiki/Repeating_decimal) [https://en.wikipedia.org/wiki/Division\_by\_zero](https://en.wikipedia.org/wiki/Division_by_zero) [https://en.wikipedia.org/wiki/Zero\_divisor](https://en.wikipedia.org/wiki/Zero_divisor)

[http://www.math.wichita.edu/history/topics/arithmetic.html#div](http://www.math.wichita.edu/history/topics/arithmetic.html#div)
