**Addition** is an elementary binary operation, written \(a + b\) (pronounced "\(a\) plus \(b\)"). It can be informally defined as the total number of objects when \(a\) objects are combined with \(b\) more. Formally, it means the cardinality of a set formed by the union of two disjoint sets with cardinalities \(a\) and \(b\). \(a\) and \(b\) are called the **summands**, and \(a + b\) is called the **sum**.

In googology, it is the first hyper operator, and forms the basis of all following hyper operators.

Addition is commutative: \(a + b = b + a\) for all values of \(a\) and \(b\). It is also associative, meaning that \((a + b) + c = a + (b + c)\). Repeated addition is called multiplication.

Zero is the additive identity, meaning that \(0 + n = n\) for all \(n\).

### In other notations Edit

Notation | Representation |
---|---|

Up-arrow notation | \(a \uparrow^{-1} b\) |

Fast-growing hierarchy | \(f_0^m(n)\) |

Hardy hierarchy | \(H_{m}(n)\) |

Slow-growing hierarchy | \(g_{\omega+m}(n)\) |