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0 (zero) is an integer representing a quantity amounting to nothing. It is the additive identity, meaning that \(a = a + 0\) for all \(a\) in various number systems such as the set of natural numbers.

Other English words for zero are nought (found mostly in the UK), nil, null, cipher (obsolete)[1], and the slang terms goose egg[2], nada, zip[3], and zilch[4]. Its ordinal form is written "0th" or "zeroth", or very rarely "noughth"[5]; these are rarely encountered except in mathematics and computer science where sequence indices can start at zero.

Properties[]

0 is an even number[6], and it is 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 exponentiated to zero \(0^0\) can either be 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, etc. to zero is one: \(a \uparrow\uparrow ...\uparrow\uparrow 0 = 1\). 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.

Trivia[]

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 \(n! = \Gamma(n + 1)\), where \(\Gamma\) is Gamma function.


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[]

Sources[]

  1. Cipher in Vocabulary.com. Retrieved 2023-02-11.
  2. Goose Egg in Marriam-Webster. Retrieved 2023-02-11.
  3. Nada in Dictionary.com. Retrieved 2023-02-11.
  4. Zilch in Vocabulary.com. Retrieved 2023-02-11.
  5. Noughth in Yourdictionry. Retrieved 2023-02-11.
  6. Robert C. Penner, Lemma B.2.2, "The integer 0 is even and is not odd", Discrete Mathematics: Proof Techniques and Mathematical Structures (1999), p. 34. ISBN 978-981-02-4088-2.
  7. Plain'N'Simple, Proof that Rayo(n) is 0 for n less than 10, Googology Wiki user blog, 2020.
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