# Real Addition is Associative

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## Theorem

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

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

## Proof

From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.

Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}, z = \eqclass {\sequence {z_n} } {}$, where $\eqclass {\sequence {x_n} } {}$, $\eqclass {\sequence {y_n} } {}$ and $\eqclass {\sequence {z_n} } {}$ are such equivalence classes.

From the definition of real addition, $x + y$ is defined as $\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$.

Thus we have:

\(\displaystyle x + \paren {y + z}\) | \(=\) | \(\displaystyle \eqclass {\sequence {x_n} } {} + \paren {\eqclass {\sequence {y_n} } {} + \eqclass {\sequence {z_n} } {} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n + z_n} } {}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {\sequence {x_n + \paren {y_n + z_n} } } {}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {\sequence {\paren {x_n + y_n} + z_n} } {}\) | Rational Addition is Associative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {\sequence {x_n + y_n} } {} + \eqclass {\sequence {z_n} } {}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} } + \eqclass {\sequence {z_n} } {}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {x + y} + z\) |

$\blacksquare$

## Sources

- 1957: Tom M. Apostol:
*Mathematical Analysis*... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-2}$ Arithmetical properties of real numbers: Axiom $2$ - 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields