Definition:Complex Number

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Definition

Informal Definition

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


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:

$\left({x_1, y_1}\right) \left({x_2, y_2}\right) := \left({x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}\right)$


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


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.


Real and Imaginary Parts

Real Part

Let $z = a + i b$ be a complex number.

The real part of $z$ is the coefficient $a$.

The real part of a complex number $z$ is usually denoted:

$\Re \left({z}\right)$
$\operatorname{Re} \left({z}\right)$
$\operatorname{re} \left({z}\right)$

or a similar variant.


Imaginary Part

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:

$\Im \left({z}\right)$
$\operatorname{Im} \left({z}\right)$
$\operatorname{im} \left({z}\right)$

or a similar variant.


Imaginary Unit

The entity $i := 0 + 1 i$ is known as the imaginary unit.


Wholly Real

A complex number $z = a + i b$ is wholly real iff $b = 0$.


Wholly Imaginary

A complex number $z = a + i b$ is wholly imaginary iff $a = 0$.


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 $\R^2$:


ComplexPlane.png


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 points on the $y$-axis.


Polar Form

For any complex number $z = x + i y \ne 0$, let:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle r\) \(=\) \(\displaystyle \) \(\displaystyle \left\vert z \right\vert = \sqrt {x^2 + y^2}\) \(\displaystyle \) \(\displaystyle \)          the modulus of $z$, and          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \theta\) \(=\) \(\displaystyle \) \(\displaystyle \arg \left({z}\right)\) \(\displaystyle \) \(\displaystyle \)          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 $\left \langle {0, 0} \right \rangle$.


Also denoted as

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


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 is usually 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.


Also see

The $a + i b$ notation usually proves more convenient; the ordered pair version is generally used only for the formal definition as given above.

  • Results about complex numbers can be found here.


Sources