The rounding and congruence functions have a long history that is closely related to the history of number theory. Many calculations use rounding of the floatingpoint and rational numbers to the closest smaller or larger integers. J. Nemorarius (1237) was one of the first mathematicians to use the quotient of two numbers and in a modern sense, but the word quotient appeared for the first time around 1250 in the writings of Meister Gernadus.
Special notations for rounding and congruence functions were introduced much later. C. F. Gauss (1801) suggested the symbol mod () for the notation of the property that the ratio is an integer. He observed that and are the congruent modulo. The number is called modulus.
C. F. Gauss (1808) and J. Liouville (1838) widely used the floor and round functions in their investigations. They and other mathematicians used different and sometimes confusing notations for those functions. The modern notations of and for floor and ceiling functions, respectively, were suggested by K. E. Iverson (1962). The notation for the rounding function was proposed by J. Hastad (1988).
