division

Division

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/Divisor https://en.wikipedia.org/wiki/Quotient https://en.wikipedia.org/wiki/Remainder https://en.wikipedia.org/wiki/Quotition_and_partition https://en.wikipedia.org/wiki/Aliquot_sequence https://en.wikipedia.org/wiki/Aliquot_sum https://en.wikipedia.org/wiki/Division_sign https://en.wikipedia.org/wiki/Euclidean_division https://en.wikipedia.org/wiki/Division_algorithm https://en.wikipedia.org/wiki/Greatest_common_divisor https://en.wikipedia.org/wiki/Euclidean_algorithm https://en.wikipedia.org/wiki/Long_division https://en.wikipedia.org/wiki/Modular_arithmetic https://en.wikipedia.org/wiki/Modulo_operation https://en.wikipedia.org/wiki/Rod_calculus https://en.wikipedia.org/wiki/Division_by_two https://en.wikipedia.org/wiki/Galley_division https://en.wikipedia.org/wiki/Inverse_element https://en.wikipedia.org/wiki/Order_of_operations https://en.wikipedia.org/wiki/Repeating_decimal https://en.wikipedia.org/wiki/Division_by_zero https://en.wikipedia.org/wiki/Zero_divisor

http://www.math.wichita.edu/history/topics/arithmetic.html#div