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Let . The **integer part function** ^{1} is defined as the largest integer less or equal to , formally it is defined as . There are many notations used for this important function but none was generally adopted:

- Gauß introduced [1] , p. 5 the bracket notation (the so-called
**Gauß bracket**) , a notation often used in number theory - Legendre [2] , p. 10,
( ) denoted it by taking from the French
*entier* - K.E.Iverson [3] p. 12, introduced the half-bracket notation and called it the
**floor function** - in programming languages this function is often denoted by .

This function can also appear also in another form, as the truncation function, where we discard the noninteger part of a positive real number. In general, the term truncation is used for reducing the number of digits right of the decimal point. Given a positive real number to be truncated and , the number of digits to be kept behind the decimal point, the truncated value is given by

For negative real numbers truncation rounds toward zero.

If and then

- , more generally for
- (the so-called
**multiplicative formula**discovered by Ch. Hermite) - , where is the divisor function

When Iverson introduced his half-bracket notation for the integer part function, he also introduced very closely related **ceiling function** (previously often called **upper integer part function**)

This function is also often named as the **post-office function**, because of the rounding the intermediate weights up to the next scale point for postal charges.

If and then

This is actually a companion function to the integer part function, and it is usually defined as the difference between and , that is . The symbol is used for this function, especially in number theory, even if the confusion with the set theoretic meaning of the same symbol is possible.

Clearly, is a periodic function with period 1 has the following Fourier expansion

(1) |

If then we have , but if then this relation is not longer true. To save the previous relation also for negative real numbers, the fractional part function is sometimes also defined by

The **nearest integer function **is formally defined as the closed integer to . Since this definition is not unambiguous for half-integers, the additional rule is necessary to adopt, for instance

- to round up by taking , or
- to round down by taking , or
- to round to even numbers in order to avoid statistical biasing.

Let us take for the definition the first possibility, that is the nearest integer function denoted by is defined by . Then

- , that is with equality if and only if

This function is defined by . It is a periodic function with period 1 and Fourier expansion

(2) |

^{1} | Also called entier function or greatest integer function or floor function |

[1] | Gauß, C. F. (1808, Jan.). Theorematis arithmetici demonstratio nova. Comment. Soc. regiae sci. Göttingen , XVI, 1-8 (Werke II, p. 1-8 ). |

[2] | Legendre, A. M. (1808). Théorie des nombres (ed. 2). |

[3] | Iverson, K. E. (1962). A Programming Language. New York: Wiley. |

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