Nearest Integer Function

The nearest integer function, also called nint or the round function, is defined such that  is the integer closest to . While the notation  is sometimes used to denote the nearest integer function (Hastad et al. 1988), this notation is rather cumbersome and is not recommended. Also note that while  is sometimes used to denote the nearest integer function,  is also commonly used to denote the floor function  (including by Gauss in his third proof of quadratic reciprocity in 1808), so this notational use is also discouraged.

Since the definition is ambiguous for half-integers, the additional rule that half-integers are always rounded to even numbers is usually added in order to avoid statistical biasing. For example, , , , , etc. This convention is followed in the C math.h library function rint, as well as in the Wolfram Language, where the nearest integer function is implemented as Round[x].

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]

The plots above illustrate  for small .

Min   Max       Re       Im

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

REFERENCES:

Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Integer Functions." Ch. 3 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 67-101, 1994.

Hastad, J.; Just, B.; Lagarias, J. C.; and Schnorr, C. P. "Polynomial Time Algorithms for Finding Integer Relations among Real Numbers." SIAM J. Comput. 18, 859-881, 1988.

Spanier, J.; Myland, J.; and Oldham, K. B. An Atlas of Functions, 2nd ed. Washington, DC: Hemisphere, 1987.