Complex Numbers under Addition form Monoid

Theorem

The set of complex numbers under addition $\left({\C, +}\right)$ forms a monoid.

Proof

Taking the monoid axioms in turn:

Monoid Axiom $\text S 0$: Closure

Complex Addition is Closed.

$\Box$

Monoid Axiom $\text S 1$: Associativity

Complex Addition is Associative.

$\Box$

Monoid Axiom $\text S 2$: Identity

From Complex Addition Identity is Zero, we have that the identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$:

$\paren {x + i y} + \paren {0 + 0 i} = \paren {x + 0} + i \paren {y + 0} = x + i y$

and similarly for $\paren {0 + 0 i} + \paren {x + i y}$.

$\Box$

Hence the result.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids