# Arithmetic Addition is Associative

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## Contents

## Theorem

Let $\mathbb F$ be one of the standard number sets: $\N, \Z, \Q, \R$ and $\C$.

Then:

- $\forall x, y, z \in \mathbb F: x + \left({y + z}\right) = \left({x + y}\right) + z$

That is, the operation of addition on the standard number sets is associative.

### Natural Number Addition is Associative

The operation of addition on the set of natural numbers $\N$ is associative:

- $\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$

### Integer Addition is Associative

The operation of addition on the set of integers $\Z$ is associative:

- $\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$

### Rational Addition is Associative

The operation of addition on the set of rational numbers $\Q$ is associative:

- $\forall x, y, z \in \Q: x + \paren {y + z} = \paren {x + y} + z$

### Real Addition is Associative

The operation of addition on the set of real numbers $\R$ is associative:

- $\forall x, y, z \in \R: x + \paren {y + z} = \paren {x + y} + z$

### Complex Addition is Associative

The operation of addition on the set of complex numbers $\C$ is associative:

- $\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$