Imaginary Part as Mapping is Surjection

Theorem

Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as:

$\forall z \in \C: \map f z = \map \Im z$

where $\map \Im z$ denotes the imaginary part of $z$.

Then $f$ is a surjection.

Proof

Let $y \in \R$ be a real number.

Let $x \in \R$ be an arbitrary real number.

Let $z \in \C$ be the complex number defined as:

$z = x + i y$

Then we have:

$\map \Im z = y$

That is:

$\exists z \in \C: \map f z = y$

The result follows by definition of surjection.

$\blacksquare$

Also see

• Real Part as Mapping is Surjection

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Direct Products