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 lớn be nonzero. A root of a function

is also sometimes known as "a zero of
."

The Schoolhouse Rock segment "My Hero, Zero" extols the virtues of zero with such praises as, "My hero, zero Such a funny little anh hùng But till you came along We counted on our fingers and toes Now you"re here khổng lồ stay & nobody really knows How wonderful you are Why we could never reach a star Without you, zero, my anh hùng How wonderful you are."

Zero is commonly taken khổng lồ have the factorization

(e.g., in the tuyetdenbatngo.com Language"s FactorInteger command). On the other hand, the divisors & divisor function
are generally taken to lớn be undefined, since by convention,
(i.e.,
divides 0) for every
except zero.

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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 lớn the power 0 is definedto be 1, which follows from the limit

0)a^x=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 và 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 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. " />

However, the limit does not exist for general complex values of

. Therefore, the choice of definition for
is usually defined 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 " />

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 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 khổng lồ
.

 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 ALSO: 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
REFERENCES:

Beeler, M. And Gosper, R.W. Sản phẩm 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. & Moll, V.H. "On Some Definite Integrals Involving the Hurwitz Zeta Function." https://arxiv.org/abs/math/0012078. 11 Dec 2000.

Finch, S.R. MathematicalConstants. 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 cảnh báo on Definite Integrals Involving Trigonometric Functions." Unpublished manuscript, n.d.

Pappas, T. "Zero-Where và When." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p.162, 1989.

Schoolhouse Rock Site. "Multiplication Rock: 0 Zero My Hero." http://www.school-house-rock.com/0.html.

Sloane, N.J.A. Sequence A007377/M0485in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious & Interesting Numbers. Middlesex, England: Penguin Books, pp.23-26, 1986.

CITE THIS AS:

Weisstein, Eric W. "Zero." From tuyetdenbatngo.com--Atuyetdenbatngo.com website Resource. Https://tuyetdenbatngo.com/Zero.html

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