0

0 (zero) is an integer representing a quantity amounting to nothing. It is the additive identity, meaning that \(a = a + 0\) for all \(a\).

Other English words for zero are nought (found mostly in the UK), nil, null, cipher (obsolete), and the slang terms goose egg, nada, zip, and zilch. Its ordinal form is written "0th", "zeroth", or very rarely "noughth"; these are rarely encountered except in mathematics and computer science where sequence indices can start at zero.

Properties
0 is an number, and neither composite nor prime since it has no prime factorization.

A number greater than zero is positive, and a number less than zero is negative. By these, 0 is neither positive nor negative.

Any number multiplied by zero equals zero: \(a \times 0 = 0\). Consequentially, \(0/a = 0\) for all \(a \not= 0\), and \(a/0\) ( by zero) is undefined.

Any number exponentiated to zero is one: \(a^0 = 1\). Zero exponentiated to any number is zero: \(0^a = 0\) Zero to the power of zero \(0^0\) can be either zero or one depending on the context. It is usually considered to be undefined, but in some cases deciding on a value can be useful.

Any number tetrated, pentated, ... to zero is one: \(a \uparrow\uparrow\ldots\uparrow\uparrow 0 = 1\). Putting zero in the left argument of a hyper operator is problematic since it creates a power tower of zeroes: \(0 \uparrow\uparrow 3 = 0^{0^0}\). Setting \(0^0 = 0\) is a little more consistent, since we get \(0^{0^0} = 0^0 = 0\), whereas \(0^0 = 1\) gives us alternating zeroes and ones: \(0^{0^0} = 0^1 = 0\) and \(0^0 = 1\). The reverse holds for the lower hyper-operators.

0! is equal to 1. This is because there is only one way to arrange zero objects — that is, doing nothing. This is compatible with many laws involving factorials, such as \(n! = \Gamma(n + 1)\).

In googology
Like 1, 0 has often been used as the default entry for googological functions. For example, most formulations of the Ackermann function allow for a base value of 0. In Bowers' Exploding Array Notation commas act as zero-dimensional separators.

Googological functions returning 0

 * Goodstein function: \(G(0)=0\)
 * Weak Goodstein function: \(g(0)=0\)
 * Kirby-Paris hydra: \(\text{Hydra}(0)=0\)
 * Buchholz hydra: \(\text{BH}(1)=0\)
 * Exploding Tree function: \(E(0)=0\)
 * Laver table: \(q(1)=0\)