Complex Addition is Commutative

Theorem

The operation of addition on the set of complex numbers is commutative:

$\forall z, w \in \C: z + w = w + z$

Proof

From the definition of complex numbers, we define the following:

\(\ds z\)

\(:=\)

\(\ds \tuple {x_1, y_1}\)

\(\ds w\)

\(:=\)

\(\ds \tuple {x_2, y_2}\)

where $x_1, x_2, y_1, y_2 \in \R$.

Then:

\(\ds z + w\)

\(=\)

\(\ds \tuple {x_1, y_1} + \tuple {x_2, y_2}\)

Definition 2 of Complex Number

\(\ds \)

\(=\)

\(\ds \tuple {x_1 + x_2, y_1 + y_2}\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds \tuple {x_2 + x_1, y_2 + y_1}\)

Real Addition is Commutative

\(\ds \)

\(=\)

\(\ds \tuple {x_2, y_2} + \tuple {x_1, y_1}\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds w + z\)

Definition 2 of Complex Number

$\blacksquare$

Examples

Example: $\paren {3 + 2 i} + \paren {-7 - i} = \paren {-7 - i} + \paren {3 + 2 i}$

Example: $\paren {3 + 2 i} + \paren {-7 - i}$

$\paren {3 + 2 i} + \paren {-7 - i} = -4 + i$

Example: $\paren {-7 - i} + \paren {3 + 2 i}$

$\paren {-7 - i} + \paren {3 + 2 i} = -4 + i$

As can be seen:

$\paren {3 + 2 i} + \paren {-7 - i} = \paren {-7 - i} + \paren {3 + 2 i}$

$\blacksquare$

Also see

• Complex Addition is Associative

Sources

• 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: $\text {(iii)}$ The fundamental operations $\text {(a)}$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Direct Products
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $2$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Axiomatic Foundations of Complex Numbers: $76 \ \text{(a)}$
• 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