Definition:Complex Number/Imaginary Unit
Definition
Let $\C = \set {a + b i: a, b \in \R}$ be the set of complex numbers.
The entity $i := 0 + 1 i$ is known as the imaginary unit.
Also denoted as
The non-italicized $\mathrm i$ can also be seen.
Using the formal definition of complex numbers, the imaginary unit is the ordered pair $\tuple {0, 1}$.
When mathematics is applied to engineering, in particular electrical and electronic engineering, the symbol $j$ is usually used
This is because $i$ is the standard symbol used to denote the flow of electric current, and to use it also for $\sqrt {-1}$ would cause untold confusion.
In some mathematical traditions, the Greek symbol $\iota$ (iota) is used for $i$.
Also see
- Results about the imaginary unit can be found here.
Historical Note
The symbol $i$ that is in widespread use for the imaginary unit was at least partly due to Leonhard Paul Euler's influence.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems
- 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
- 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.1$ The Complex Field $C$
- 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