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 power nguồn 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  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 khổng lồ be indeterminate.However, defining  allows some formulas khổng lồ 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 and Moll 2000), where =b>c" />, , 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 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 to 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, ...
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| 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, Zero-Form, Zero Matrix, Zero-Sum Game, ZerofreeExplore with conhantaohpg.com|Alpha
ReferencesBeeler, M. Và Gosper, R.W. Thành tựu 57 in Beeler, M.; Gosper, R.W.; và 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. & 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 chú ý 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 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 On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious và Interesting Numbers. Middlesex, England: Penguin Books, pp.23-26, 1986.YouTube. "My Hero, Zero." https://www.youtube.com/watch?v=L1yPuSP1HPY.
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Referenced on conhantaohpg.com|AlphaZeroCite this as:Weisstein, Eric W. "Zero." From conhantaohpg.com--Aconhantaohpg.com website Resource. Https://conhantaohpg.com/Zero.html Subject classificationsMore...Less... Created, developed & nurtured by Eric Weisstein at conhantaohpg.com Research
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