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Nearest Integer Function

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The nearest integer function (also called nint or x-rounded) g(x) = {x} assigns the nearest integer to x for every real number. If x is in the middle of two integers, the function returns the largest of the two numbers (Gerstein, 2012), which avoids statistical bias (Nemati et al., 2013).

nearest integer function

Distance to the nearest integer function || . ||


A few examples:

  • {1.99} = 2,
  • {6} = 6,
  • {3.5} = 4,
  • {-0.6} = -1,
  • {-4.5} = -4.

The function is mostly used in number theory and approximation theory, with some application in dynamical system theory. It can also “tidy up a good many otherwise complicated formulas” such as the formula for the number of permutations of n letters with no fixed points:


Which, when you realize that the sum is a truncation of the infinite series for e-1, simplifies to:
Dn = ||n!/e|| for n ≥ 1.
(Wilf, 1987, p. 855.).


Similar functions that belong to the same family — functions that return real integers based on a particular rule — include the ceiling (least integer) function and floor (greatest integer) function.

Notation for the Nearest Integer Function

The nearest integer function doesn’t have a commonly accepted standard notation. While it’s common to use curly brackets to represent the nearest integer function (as in the examples above), the symbol || || is also used; For example, ||0.49|| = 0 (Brown, 1998). Other notation includes the symbol “(x)” (Singh, 2021) and (Wilf, 1987).

Occasionally, brackets [] are used, but these can be confused with equivalence class. In addition, the floor function is sometimes denoted with brackets, especially in older texts, compounding the potential confusion. The notation |_x] is also sometimes used (Hastad et al. 1988), but this notation is cumbersome and not recommended (Nemati et al, 2013).

References

Brown, P. (1998). Advances in Chromatography – Volume 39 – Page 154. Taylor & Francis.
Gerstein, L. (2012). Introduction to Mathematical Structures and Proofs. Springer.
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.
Nemati et al., (2013). Using Black Holes Algorithm in Discrete Space by Nearest
Integer Function. IAES International Journal of Artificial Intelligence (IJ-AI)
Vol. 2, No. 4, December 2013, pp. 173~178.
Singh, S. (2021). 3.9 Greatest and least integer functions. Retrieved January 27, 2021 from: https://cnx.org/contents/[email protected]:[email protected]/Greatest-and-least-integer-functions
Wilf, H. (1987). The Editor’s Corner: Strings, Substrings, and the `Nearest Integer’ Function. The American Mathematical Monthly Vol. 94, No. 9 (Nov., 1987), pp. 855-860 (6 pages) Published By: Taylor & Francis, Ltd.

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Stephanie Glen. "Nearest Integer Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/nearest-integer-function/
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