The fractional part of a real number is the non-integer part of that number, which can be thought of as the difference between the number and the largest integer less than or equal to that number (its floor).
Articles by others on the same topic
The fractional part of a real number isIt is the amount left after removing the lower enclosing integer, even for negative inputs. Thus . It is not the signed string of digits obtained by truncation toward zero.
Translation by an integer leaves the fractional part unchanged: . It is therefore a periodic function with period one, increasing from zero toward one on each unit interval, then returning to zero at the next integer. In Euclidean division with , ; for , this is .
The notation is useful for following repeated motion modulo an integer. In normality and equidistribution under integer multiplication, the values shift the digits of a positional notation; the fractional part discards the leading integer portion without losing the remaining digits. Its braces must be distinguished by context from the braces denoting a set.