Inverse for Complex Addition

Theorem

Let $z = x + i y \in \C$ be a complex number.

Let $-z = -x - i y \in \C$ be the negative of $z$.

Then $-z$ is the inverse element of $z$ under the operation of complex addition:

$\forall z \in \C: \exists -z \in \C: z + \paren {-z} = 0 = \paren {-z} + z$

Proof

From Complex Addition Identity is Zero, the identity element for $\struct {\C, +}$ is $0 + 0 i$.

Then:

\(\ds \)

\(\)

\(\ds \paren {x + i y} + \paren {-x - i y}\)

\(\ds \)

\(=\)

\(\ds \paren {x - x} + i \paren {y - y}\)

\(\ds \)

\(=\)

\(\ds 0 + 0 i\)

Similarly for $\paren {-x - i y} + \paren {x + i y}$.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $8$
• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.2$ The algebraic structure of the complex numbers