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The floor of x is also called the integral part, integer part, greatest integer, or entier of x, and was historically denoted [x] (among other notations). [2] However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers. For n an integer, ⌊n⌋ = ⌈n⌉ = n.
Partitions of n with largest part k. In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.)
The word integer comes from the Latin integer meaning "whole" or (literally) "untouched", from in ("not") plus tangere ("to touch"). "Entire" derives from the same origin via the French word entier, which means both entire and integer. [9] Historically the term was used for a number that was a multiple of 1, [10] [11] or to the whole part of a ...
Split into its integer part and fractional part, = ⌊ ⌋ + {}.There is exactly one ′ {, …,} with ⌊ ⌋ = ⌊ + ′ ⌋ < ⌊ + ′ ⌋ = ⌊ ⌋ + By subtracting the same integer ⌊ ⌋ from inside the floor operations on the left and right sides of this inequality, it may be rewritten as
The part from the decimal separator to the right is the fractional part, which equals the difference between the numeral and its integer part. When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example, .1234, instead of 0.1234). In normal writing, this is generally ...
The fractional part or decimal part [1] of a non‐negative real number is the excess beyond that number's integer part. The latter is defined as the largest integer not greater than x , called floor of x or ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } .
For instance, the following algorithmic procedure will give the standard decimal representation: Given , we first define (the integer part of ) to be the largest integer such that (i.e., = ⌊ ⌋). If x = a 0 {\displaystyle x=a_{0}} the procedure terminates.
It has two definitions: either the integer part of a division (in the case of Euclidean division) [2] or a fraction or ratio (in the case of a general division). For example, when dividing 20 (the dividend ) by 3 (the divisor ), the quotient is 6 (with a remainder of 2) in the first sense and 6 2 3 = 6.66... {\displaystyle 6{\tfrac {2}{3}}=6.66 ...