Definition:Complex Number/Definition 1

Definition

A complex number is a number in the form $a + b i$ or $a + i b$ where:

$a$ and $b$ are real numbers

$i$ is a square root of $-1$, that is, $i = \sqrt {-1}$.

Notation

The set of complex numbers is usually denoted $\C$.

Variants on $\C$ are often seen, for example $\mathbf C$, $\CC$ and $\mathfrak C$, or even just $C$.

When $a$ and $b$ are symbols representing variables or constants, the form $a + i b$ is usually (but not universally) seen.

Similarly, when $a$ and $b$ are actual numbers, for example $3$ and $4$, it is usually (but not universally) written $3 + 4 i$.

Also see

• Equivalence of Definitions of Complex Number

• Results about complex numbers can be found here.

Sources

• 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers
• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Cartesian Form: $3.7.1$
• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 6$: Complex Numbers: Definitions Involving Complex Numbers
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 1$. Introduction
• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Complex Number System
• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.1$ Complex numbers and their representation
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): complex number
• 2003: John H. Conway and Derek A. Smith: On Quaternions And Octonions ... (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): 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: Definitions Involving Complex Numbers