Definition:Complex Number/Imaginary Part
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Definition
Let $z = a + i b$ be a complex number.
The imaginary part of $z$ is the coefficient $b$ (note: not $i b$).
The imaginary part of a complex number $z$ is usually denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\map \Im z$ or $\Im z$.
Polar Form
Let $z$ be a complex number expressed in polar form:
- $z = \polar {r, \theta}$
The imaginary part of $z$ is:
- $\map \Im z = r \sin \theta$
Also denoted as
Variants of $\map \Im z$ for the imaginary part of a complex number $z$ are:
- $\map {\mathrm {Im} } z$
- $\mathrm {Im} \set z$
- $\map {\mathscr I} z$
- $\map {\mathrm {im} } z$
- $\map {\mathfrak I} z$
- $\map {\mathbf I} z$ or $\mathbf I z$
While the fraktur font is falling out of fashion, because of its cumbersome appearance and difficulty to render in longhand, its use for this application is conveniently unambiguous.
Also see
- Results about imaginary parts can be found here.
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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Polar Form: $3.7.6$: Imaginary Part
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Example $12.3$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 6$: Complex Numbers: Definitions Involving Complex Numbers
- 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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): imaginary part
- 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.1$ The Complex Field $C$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): complex number
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): imaginary part
- 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): imaginary part