Definitions of the rounding and congruence functions

The rounding and congruence functions include seven basic functions. They all deal with the separation of integer or fractional parts from real and complex numbers: the floor function (entire part function) , the nearest integer function (round) , the ceiling function (least integer) , the integer part , the fractional part , the modulo function (congruence) , and the integer part of the quotient (quotient or integer division) .

The floor function (entire function) can be considered as the basic function of this group. The other six functions can be uniquely defined through the floor function.

Floor

For real , the floor function is the greatest integer less than or equal to .

For arbitrary complex , the function can be described (or defined) by the following formulas:

Examples: , , , , , ,.

Round

For real , the rounding function is the integer closest to (if ).

For arbitrary , the round function can be described (or defined) by the following formulas:

Examples: , , , , , , , .

Ceiling

For real , the ceiling function is the smallest integer greater than or equal to .

For arbitrary , the function can be described (or defined) by the following formulas:

Examples: , , , , , , , .

Integer part

For real , the function integer part is the integer part of .

For arbitrary , the function can be described (or defined) by the following formulas:

Examples: , , , , , , ,.

Fractional part

For real , the function fractional part is the fractional part of .

For arbitrary , the function can be described (or defined) by the following formulas:

Examples: , , , , , , ,.

Mod

For complex and , the mod function is the remainder of the division of by . The sign of for real , is always the same as the sign of .

The mod function can be described (or defined) by the following formula:

The functional property makes the behavior of similar to the behavior of .

Examples: , , , , , .

Quotient

For complex and , the integer part of the quotient (quotient) function is the integer quotient of and .

The quotient function can be described (or defined) by the following formula:

Examples: , , , , , .