division
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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.
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