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
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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): complex number