“Zero”的概念、定义、习题解题方法及翻译-数学辞典

The Integer denoted 0 which, when used as a counting number, means that no objects are present. It is the onlyInteger (and, in fact, the only Real Number) which is neither Negative nor Positive. A number which is not zero is said to be Nonzero.

Because the number of Permutations of 0 elements is 1, 0! (zero Factorial) is often defined as1. This definition is useful in expressing many mathematical identities in simple form. A number other than 0 takento the Power 0 is defined to be 1.

is undefined, but defining

allows concise statement of the beautifulanalytical formula for the integral of the generalized Sinc Function

The following table gives the first few numbers

such that teh decimal expansion of

contains no zeros, for small

. The largest known

for which

contain no zeros is 86 (Madachy 1979), with no other

(M. Cook),improving the

limit obtained by Beeler et al. (1972). The values

such that the positions of theright-most zero in

increases are 10, 20, 30, 40, 46, 68, 93, 95, 129, 176, 229, 700, 1757, 1958, 7931, 57356,269518, ... (Sloane's A031140). The positions in which the right-most zeros occur are 2, 5, 8, 11, 12, 13, 14, 23, 36, 38,54, 57, 59, 93, 115, 119, 120, 121, 136, 138, 164, ... (Sloane's A031141). The right-most zero of

occurs atthe 217th decimal place, the farthest over for powers up to

	Sloane	such that contains no 0s
2	Sloane's A007377	1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24, 25, 27, 28, ...
3	Sloane's A030700	1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 19, 23, 24, 26, 27, 28, ...
4	Sloane's A030701	1, 2, 3, 4, 7, 8, 9, 12, 14, 16, 17, 18, 36, 38, 43, ...
5	Sloane's A008839	1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58, ...
6	Sloane's A030702	1, 2, 3, 4, 5, 6, 7, 8, 12, 17, 24, 29, 44, ...
7	Sloane's A030703	1, 2, 3, 6, 7, 10, 11, 19, 35
8	Sloane's A030704	1, 2, 3, 5, 6, 8, 9, 11, 12, 13, 17, 24, 27
9	Sloane's A030705	1, 2, 3, 4, 6, 7, 12, 13, 14, 17, 34
11	Sloane's 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 probabilitythat 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, ... (Sloane's A020665).

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 57, Feb. 1972.

Kogan, S. ``A Note on Definite Integrals Involving Trigonometric Functions.'' http://www.mathsoft.com/asolve/constant/pi/sin/sin.html.

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. SequenceA007377/M0485in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.