0)a^x=1. " />  (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 1" /> gives smaller & smaller numbers that approach one from above, while doing the same with a number between 0 và 1 gives larger & larger numbers that approach one from below. For square roots, the total nguồn taken is , which approaches 0 as is large, giving in the limit that is large. itself is undefined. The lack of a welldefined meaning for this quantity follows from the mutually contradictory facts that is always 1, so should equal 1, but is always 0 (for 0" />), so should equal 0. It could be argued that is a natural definition since
0)n^n=lim_(n>0^+)n^n=lim_(n>0^)n^n=1. " />  (2)
However, the limit does not exist for general complex values of . Therefore, the choice of definition for is usually defined to lớn be indeterminate.However, defining allows some formulas to lớn 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
^c " />  (3)
given by Kogan (cf. Espinosa và Moll 2000), where =b>c" />, , và 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 lớn 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 occurs at the 217th decimal place, the farthest over for powers up lớn .
 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).
See also1, 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, ZeroForm, Zero Matrix, ZeroSum Game, ZerofreeExplore with romanhords.comAlpha
ReferencesBeeler, M. & Gosper, R.W. Chiến thắng 57 in Beeler, M.; Gosper, R.W.; & Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239, 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, 403422, 1992.Knuth, D.E. The Art of Computer Programming, Vol.1: Fundamental Algorithms, 3rd ed. Reading, MA: AddisonWesley, p.57, 1997.Kogan, S. "A chú ý on Definite Integrals Involving Trigonometric Functions." Unpublished manuscript, n.d.Madachy, J.S. Madachy"s Mathematical Recreations. New York: Dover, pp.127128, 1979.Pappas, T. "ZeroWhere và When." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p.162, 1989.Sloane, N.J.A. Sequence A007377/M0485 in "The OnLine Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp.2326, 1986.YouTube. "My Hero, Zero." https://www.youtube.com/watch?v=L1yPuSP1HPY.
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