 TOPICS # Zero

Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. It is the only integer (and, in fact, the only real number) that is neither negative nor positive. A number which is not zero is said to be nonzero. A root of a function is also sometimes known as "a zero of ." The Schoolhouse Rock segment "My Hero, Zero" (Multiplication Rock, Season 1, Episode 2, 1973) extols the virtues of zero with such praises as, "My hero, zero Such a funny little hero But till you came along We counted on our fingers and toes Now you're here to stay And nobody really knows How wonderful you are Why we could never reach a star Without you, zero, my hero How wonderful you are."

Zero is commonly taken to have the factorization (e.g., in the Wolfram Language's FactorInteger[n] command). On the other hand, the divisors and divisor function are generally taken to be undefined, since by convention, (i.e., divides 0) for every except zero.

Because the number of permutations of 0 elements is 1, (zero factorial) is defined as 1 (Wells 1986, p. 31). This definition is useful in expressing many mathematical identities in simple form. A number other than 0 taken to the power 0 is defined to be 1, which follows from the limit (1)

This fact is illustrated by the convergence of curves at in the plot above, which shows for , 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number gives smaller and smaller numbers that approach one from above, while doing the same with a number between 0 and 1 gives larger and larger numbers that approach one from below. For square roots, the total power taken is , which approaches 0 as is large, giving in the limit that is large. itself is undefined. The lack of a well-defined meaning for this quantity follows from the mutually contradictory facts that is always 1, so should equal 1, but is always 0 (for ), so should equal 0. It could be argued that is a natural definition since (2)

However, the limit does not exist for general complex values of . Therefore, the choice of definition for is usually defined to be indeterminate.

However, defining allows some formulas to be expressed simply (Knuth 1992; Knuth 1997, p. 57), an example of which is the beautiful analytical formula for the integral of the generalized sinc function (3)

given by Kogan (cf. Espinosa and Moll 2000), where , , and is the floor function.

Richardson's theorem is a fundamental result in decidability theory which establishes that the determination of whether even simple expressions are identically equal to zero is undecidable in principle, let alone in practice.

The following table gives the first few numbers such that the decimal expansion of contains no zeros for small (a problem that resembles Gelfand's question.) The largest known for which contain no zeros is 86 (Madachy 1979), with no other (M. Cook, pers. comm., Sep. 26, 1997 and Mar. 16, 1998), improving the limit obtained by Beeler and Gosper (1972). The values such that the positions of the rightmost zero in increases are 10, 20, 30, 40, 46, 68, 93, 95, 129, 176, 229, 700, 1757, 1958, 7931, 57356, 269518, ... (OEIS A031140). The positions in which the rightmost zeros occur are 2, 5, 8, 11, 12, 13, 14, 23, 36, 38, 54, 57, 59, 93, 115, 119, 120, 121, 136, 138, 164, ... (OEIS A031141). The rightmost zero of occurs at the 217th decimal place, the farthest over for powers up to . Sloane such that contains no 0s 2 A007377 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, ... 3 A030700 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, ... 4 A030701 1, 2, 3, 4, 7, 8, 9, 12, 14, 16, 17, 18, 36, 38, 43, ... 5 A008839 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58, ... 6 A030702 1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44, ... 7 A030703 1, 2, 3, 6, 7, 10, 11, 19, 35 8 A030704 1, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27 9 A030705 1, 2, 3, 4, 6, 7, 12, 13, 14, 17, 34 11 A030706 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 41, ...

While it has not been proven that the numbers listed above are the only ones without zeros for a given base, the probability that any additional ones exist is vanishingly small. Under this assumption, the sequence of largest such that contains no zeros for , 3, ... is then given by 86, 68, 43, 58, 44, 35, 27, 34, 0, 41, ... (OEIS A020665).

1, 2, 10, Airy Function Zeros, Approximate Zero, Bessel Function Zeros, Constant Problem, Division by Zero, Fallacy, Hidden Zero, Identically Zero, Naught, Negative, Nonnegative, Nonzero, Positive, Power, Richardson's Theorem, Root, Uniformity Conjecture, Vanishing, Vanish Identically, Zero Divisor, Zero-Form, Zero Matrix, Zero-Sum Game, Zerofree

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## References

Beeler, M. and Gosper, R. W. Item 57 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 22, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item57.Espinosa, O. and Moll, V. H. "On Some Definite Integrals Involving the Hurwitz Zeta Function." https://arxiv.org/abs/math/0012078. 11 Dec 2000.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Knuth, D. E. "Two Notes on Notation." Amer. Math. Monthly 99, 403-422, 1992.Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 57, 1997.Kogan, S. "A Note on Definite Integrals Involving Trigonometric Functions." Unpublished manuscript, n.d.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 127-128, 1979.Pappas, T. "Zero-Where & When." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 162, 1989.Sloane, N. J. A. Sequence A007377/M0485 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 23-26, 1986.YouTube. "My Hero, Zero." https://www.youtube.com/watch?v=L1yPuSP1HPY.

Zero

## Cite this as:

Weisstein, Eric W. "Zero." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zero.html