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" 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
![int_0^infty(sin^ax)/(x^b)dx=(pi^(1-c)(-1)^(|_(a-b)/2_|))/(2^(a-c)(b-1)!)sum_(k=0)^(|_a/2_|-c)(-1)^k(a; k)(a-2k)^(b-1)[ln(a-2k)]^c](/images/equations/Zero/NumberedEquation3.gif) |
(3)
|
given by Kogan, 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).
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