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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 $$a \uparrow^{-2} b$$
Fast-growing hierarchy $$f_0^m(n)$$
Hardy hierarchy $$H_{m}(n)$$
Slow-growing hierarchy $$g_{\omega+m}(n)$$