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< -1 | 1 > |
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Numbers 0 - 99 | |||||||||

0 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |

30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 |

40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 |

50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 |

60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 |

70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 |

80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 |

90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 |

**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 even 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 \(a/0\) (division 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 creates a power tower of zeroes: \(0 \uparrow\uparrow 3 = 0^{0^0}\). Setting \(0^0 = 1\), we obtain a sequence of alternating zeroes and ones: \(0^{0^0} = 0^1 = 0\) and \(0^0 = 1\). The reverse holds for the lower hyper-operators.

0 is the only non-negative integer that is not a natural number. Non-negative integers include 0, 1, 2, 3, 4, etc. while naturals numbers skip zero and continue: 1, 2, 3, 4, 5, etc.

0 is the smallest non-negative number.

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: \(q(1)=0\)
- Rayo's function: \(\text{Rayo}(0)=\cdots=\text{Rayo}(9)=0\)
^{[1]}

## Sources

- ↑ Plain'N'Simple, Proof that Rayo(n) is 0 for n less than 10, Googology Wiki user blog, 2020.