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 Bakhshali manuscript is among the oldest surviving manuscripts of South Asian mathematics and provides early evidence for the use of zero in computation. Written on birch bark, it uses a dot both as a placeholder in base 10 notation and as a number in arithmetic operations (Plofker et al. 2017). Its age has long been debated, and a revised radiocarbon study places its production broadly around AD 800-1100 (Chivall et al. 2024).
The Schoolhouse Rock segment "My Hero, Zero" (Multiplication Rock, Season 1, Episode 2, 1973) 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.
The value assigned to 
depends on context. In analysis it is often left undefined, and 
is an indeterminate form
of a two-variable limit. For principal complex powers,
however,
 |
(2)
|
Indeed, 
and 
for the principal complex logarithm.
This one-variable limit does not settle the two-variable
form. For example, as 
,
the paths 
and 
both approach 
but give different limiting values for 
.
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.svg) |
(3)
|
given by Kogan (1999), where 
and 
are positive
integers, 
,
, 
, and 
is the floor function.
When 
, the logarithmic factor
is interpreted as 1 and a summand with 
is taken as 0.
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 positive integers 
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).
Counting the units digit as position 0, 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 position 217,
the 218th digit from the right, 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 conjectured sequence of largest positive
integers 
such that 
contains no zeros for 
, 3, ..., using 0 when no positive exponent
works, is then given by 86, 68, 43, 58, 44, 35, 27, 34, 0, 41, ... (OEIS A020665).
