0

0 (zero) is an integer. It is the additive identity, meaning that \(a = a + 0\) for all \(a\). 0 is an number.

A number greater than zero is positive, and a number less than zero is negative.

Any number multiplied by zero equals zero: \(a \times 0 = 0\). Consequentially, \(0/a = 0\) for all \(a\), 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 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.

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 can be thought of as a rank 0 separator.