Definition:Addition/Complex Numbers

Definition

The addition operation in the domain of complex numbers $\C$ is written $+$.

Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R, i^2 = -1$.

Then $z + w$ is defined as:

$\paren {a + i b} + \paren {c + i d} = \paren {a + c} + i \paren {b + d}$

Complex Addition

When the formal definition of complex numbers is used, complex addition is defined thus:

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.

Then $+$ is a binary operation defined as:

$\tuple {x_1, y_1} + \tuple {x_2, y_2}:= \tuple {x_1 + x_2, y_1 + y_2}$

Examples

Example: $\paren {3 + 2 i} + \paren {5 + 6 i}$

$\paren {3 + 2 i} + \paren {5 + 6 i} = 8 + 8 i$

Example: $\paren {-1 + 4 i} + \paren {2 + \paren {-7} i}$

$\paren {-1 + 4 i} + \paren {2 + \paren {-7} i} = 1 + \paren {-3} i$

Also see

• Complex Addition is Commutative
• Complex Addition is Associative

• Identity Element of Addition on Numbers
• Inverse Elements for Addition on Numbers

• Definition:Complex Multiplication

• Results about complex addition can be found here.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: $(1.4)$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 6$: Complex Numbers: Addition of Complex Numbers: $6.2$
• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Fundamental Operations with Complex Numbers: $1$. Addition
• 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
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): addition
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): complex number
• 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.1$ The Complex Field $C$
• 2003: John H. Conway and Derek A. Smith: On Quaternions And Octonions ... (previous) ... (next): $\S 1$: The Complex Numbers and Their Applications to $1$- and $2$-Dimensional Geometry
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): addition
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): complex number
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 4$: Complex Numbers: Arithmetic of Complex Numbers: $4.2.$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): addition (of complex numbers)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): complex number
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): addition (of complex numbers)