Addition

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 on \(\mathcal{N}\) and \(\mathcal{R}\): \(a + b = b + a\) for all natural or real values of \(a\) and \(b\). It is also, meaning that \((a + b) + c = a + (b + c)\). Repeated addition is called multiplication.

However, addition is not commutative on ordinals. For any limit ordinal \(\alpha\), \(1+\alpha = \alpha \neq \alpha+1\).

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