Definition:Complex Number
Contents |
Informal 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 the square root of $-1$, i.e. $\sqrt {-1}$.
The set of all complex numbers is denoted $\C$.
Variants on $\C$ are often seen, for example $\mathbf C$ and $\mathcal C$, or even just $C$.
Formal Definition
A complex number is an ordered pair $\left({x, y}\right)$ where $x, y \in \R$ are real numbers, on which the operations of addition and multiplication are defined as follows:
Complex Addition
Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.
Then $\left({x_1, y_1}\right) + \left({x_2, y_2}\right)$ is defined as:
- $\left({x_1, y_1}\right) + \left({x_2, y_2}\right):= \left({x_1 + x_2, y_1 + y_2}\right)$
Complex Multiplication
Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.
Then $\left({x_1, y_1}\right) \left({x_2, y_2}\right)$ is defined as:
- $\left({x_1, y_1}\right) \left({x_2, y_2}\right) := \left({x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}\right)$
Construction from Cayley-Dickson Construction
The complex numbers can be defined by the Cayley-Dickson construction from the set of real numbers $\R$.
From Real Numbers form Algebra, $\R$ forms a nicely normed $*$-algebra.
Let $a, b \in \R$.
Then $\left({a, b}\right) \in \C$, where:
- $\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$
- $\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$
where:
- $\overline a$ is the conjugate of $a$
and
- $\overline {\left({a, b}\right)}$ is the conjugation operation on $\C$.
From Real Numbers form Algebra, $\overline a = a$ and so the above translate into:
- $\left({a, b}\right) \left({c, d}\right) = \left({a c - d b, a d + c b}\right)$
- $\overline {\left({a, b}\right)} = \left({a, -b}\right)$
It is clear by direct comparison with the formal definition that this construction genuinely does generate the complex numbers.
Imaginary Unit
The entity $i$ as defined here is called the imaginary unit.
Equivalence of Definitions
The two definitions as given above are equivalent.
The $a + i b$ notation usually proves more convenient; the ordered pair version is generally used only for the formal definition as given above.
Real Part
The real part of a complex number $a + i b$ is the coefficient $a$.
The real part of a complex number $z$ is often denoted $\Re \left({z}\right)$ or $\operatorname{Re} \left({z}\right)$ or $\operatorname{re} \left({z}\right)$.
Imaginary Part
The imaginary part of a complex number $a + i b$ is the coefficient $b$ (note: not $i b$).
The imaginary part of a complex number $z$ is often denoted $\Im \left({z}\right)$ or $\operatorname{Im} \left({z}\right)$ or $\operatorname{im} \left({z}\right)$.
Wholly Real
The complex number $z = a + i b$ is called wholly real or completely real, or entirely real, etc. iff $b = 0$.
Wholly Imaginary
The complex number $z = a + i b$ is called wholly imaginary or completely imaginary, or entirely imaginary, etc. iff $a = 0$.
Notation
When $a$ and $b$ are symbols representing variables or constants, the form $a + i b$ is usually seen.
When $a$ and $b$ are actual numbers, for example 3 and 4, it usually gets written $3 + 4 i$.
When mathematics is applied to engineering, in particular electrical and electronic engineering, the symbol $j$ is usually used, as $i$ is the standard symbol used to denote the flow of electric current, and to use it also for $\sqrt {-1}$ would cause untold confusion.
Complex Plane
Because a complex number can be expressed as an ordered pair, we can plot the number $x + i y$ on the Real Number Plane $\R^2$:
See Argand diagram.
Real Axis
Complex numbers of the form $\left({x, 0}\right)$, being wholly real, appear as points on the $x$-axis.
Imaginary Axis
Complex numbers of the form $\left({0, y}\right)$, being wholly imaginary, appear as points on the $y$-axis.
Polar Form
For any complex number $z = x + i y \ne 0$, let:
| \(\displaystyle \) | \(\displaystyle r\) | \(=\) | \(\displaystyle \left\vert z \right\vert = \sqrt {x^2 + y^2}\) | \(\displaystyle \) | the modulus of $z$, and | ||
| \(\displaystyle \) | \(\displaystyle \theta\) | \(=\) | \(\displaystyle \arg \left({z}\right)\) | \(\displaystyle \) | the argument of $z$ (the angle which $z$ yields with the real line) |
where $x, y \in \R$.
From the definition of $\arg \left({z}\right)$:
- $(1): \quad \dfrac x r = \cos \theta$
- $(2): \quad \dfrac y r = \sin \theta$
which implies that:
- $x = r \cos \theta$
- $y = r \sin \theta$
which in turn means that any number $z = x + i y \ne 0$ can be written as:
- $z = x + i y = r \left({\cos \theta + i \sin \theta}\right)$
The pair $\left \langle {r, \theta} \right \rangle$ is called the polar form of the complex number $z \ne 0$.
The number $z = 0 + 0i$ is defined as $\left \langle {0, 0} \right \rangle$.
From Euler's Theorem we have that $e^{i \theta} = \cos \theta + i \sin \theta$, so we can also write $z$ in the form:
- $z = r e^{i \theta}$
Sources
- Seth Warner: Modern Algebra (1965): $\S 1$
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.2$
- Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $6.6$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.1$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $4$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 2 \ \text{(b)}$
