Arithmetic Multiplication is Commutative

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Theorem

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

Then:

$\forall x, y \in \mathbb F: x + y = y + x$

That is, the operation of multiplication on the standard number sets is commutative.


Natural Number Multiplication is Commutative

The operation of multiplication on the set of natural numbers $\N$ is commutative:

$\forall x, y \in \N: x \times y = y \times x$


Integer Multiplication is Commutative

The operation of multiplication on the set of integers $\Z$ is commutative:

$\forall x, y \in \Z: x \times y = y \times x$


Rational Multiplication is Commutative

The operation of multiplication on the set of rational numbers $\Q$ is commutative:

$\forall x, y \in \Q: x \times y = y \times x$


Real Multiplication is Commutative

The operation of multiplication on the set of real numbers $\R$ is commutative:

$\forall x, y \in \R: x \times y = y \times x$


Complex Multiplication is Commutative

The operation of multiplication on the set of complex numbers $\C$ is commutative:

$\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$