Integer Part

The function  gives the integer part of . In many computer languages, the function is denoted int(x). It is related to the floor and ceiling functions  and  by

(1)

The integer part function satisfies

(2)

and is implemented in the Wolfram Language as IntegerPart[x]. This definition is chosen so that , where  is the fractional part. Although Spanier and Oldham (1987) use the same definition as in the Wolfram Language, they mention the formula only very briefly and then say it will not be used further. Graham et al. (1994), and perhaps most other mathematicians, use the term "integer" part interchangeably with the floor function .

Min   Max       Re       Im

The integer part function can also be extended to the complex plane, as illustrated above.

Since usage concerning fractional part/value and integer part/value can be confusing, the following table gives a summary of names and notations used. Here, S&O indicates Spanier and Oldham (1987).

notation	name	S&O	Graham et al.	Wolfram Language
	ceiling function	--	ceiling, least integer	Ceiling[x]
	congruence	--	--	Mod[m, n]
	floor function		floor, greatest integer, integer part	Floor[x]
	fractional value		fractional part or	SawtoothWave[x]
	fractional part		no name	FractionalPart[x]
	integer part		no name	IntegerPart[x]
	nearest integer function	--	--	Round[x]
	quotient	--	--	Quotient[m, n]

REFERENCES:

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 67, 1994.

Spanier, J. and Oldham, K. B. "The Integer-Value Int() and Fractional-Value frac() Functions." Ch. 9 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 71-78, 1987.