Complex Addition is Associative

Theorem

The operation of addition on the set of complex numbers $\C$ is associative:

$\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$

Proof

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

\(\ds z_1\)

\(:=\)

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

\(\ds z_2\)

\(:=\)

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

\(\ds z_3\)

\(:=\)

\(\ds \tuple {x_3, y_3}\)

where $x_1, x_2, x_3, y_1, y_2, y_3 \in \R$.

Thus:

\(\ds z_1 + \paren {z_2 + z_3}\)

\(=\)

\(\ds \tuple {x_1, y_1} + \paren {\tuple {x_2, y_2} + \tuple {x_3, y_3} }\)

Definition 2 of Complex Number

\(\ds \)

\(=\)

\(\ds \tuple {x_1, y_1} + \paren {\tuple {x_2 + x_3, y_2 + y_3} }\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds \tuple {x_1 + \paren {x_2 + x_3}, y_1 + \paren {y_2 + y_3} }\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds \tuple {\paren {x_1 + x_2} + x_3, \paren {y_1 + y_2} + y_3}\)

Real Addition is Associative

\(\ds \)

\(=\)

\(\ds \paren {\tuple {x_1 + x_2, y_1 + y_2} } + \tuple {x_3, y_3}\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds \paren {\tuple {x_1, y_1} + \tuple {x_2, y_2} } + \tuple {x_3, y_3}\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds \paren {z_1 + z_2} + z_3\)

Definition 2 of Complex Number

$\blacksquare$

Examples

Example: $\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = \paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$

Example: $\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} }$

$\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = 11$

Example: $\paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$

$\paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i} = 11$

As can be seen:

$\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = \paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$

$\blacksquare$

Also see

• Complex Addition is Commutative

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: $77 \ \text{(a)}$
• 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