Knowra Floor and ceiling functions Floor and ceiling functions The floor of a real number is the greatest integer no greater than it; the ceiling is the least integer no less than it. They round values down or up to integers, respectively.
Greatest integer function : The function that assigns each real number the greatest integer less than or equal to it. This is the floor function, one half of the paired definition.
Real number : An element of the continuous number system containing the rational and irrational numbers. Both functions accept every real number as input.
Rounding : The process of replacing a number with a nearby value at a chosen precision. Floor and ceiling are directed rounding rules, unlike rounding to nearest.
Round half to even : A rounding rule that sends exact halfway cases to the nearest even integer. It rounds to the nearest integer rather than always downward or upward.
Ceiling function : The function that assigns each real number the least integer greater than or equal to it. Its upward rounding complements the floor function's downward rounding.
Integer : A whole number, positive, negative, or zero, with no fractional part. Every floor and ceiling value is an integer.
Modulo operation : An operation that gives the remainder associated with division by a specified modulus. A floor-based quotient yields a consistent nonnegative remainder for positive moduli.
Truncation (mathematics) : A rounding operation that discards digits or fractional parts, moving toward zero in common numerical use. For negative nonintegers, truncation does not agree with the floor function.
Integer part : A truncation operation that removes a number's fractional part, often rounding toward zero. For negative nonintegers, truncation differs from floor's rounding toward negative infinity.
Order relation : A relation that compares elements using properties such as less than and greater than. The defining bounds use inequalities to select the greatest or least eligible integer.
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