Definition:Complex Number

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Definition

Complex vowels.png

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


Formal Definition

A complex number is an ordered pair $\tuple {x, y}$ where $x, y \in \R$ are real numbers, on which the operations of addition and multiplication are defined as follows:


Complex Addition

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.

Then $+$ is a binary operation defined as:

$\tuple {x_1, y_1} + \tuple {x_2, y_2}:= \tuple {x_1 + x_2, y_1 + y_2}$


Complex Multiplication

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.


Then $\tuple {x_1, y_1} \tuple {x_2, y_2}$ is defined as:

$\tuple {x_1, y_1} \tuple {x_2, y_2} := \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}$


Scalar Product

Let $\tuple {x, y}$ be a complex number.

Let $m \in \R$ be a real number.


Then $m \tuple {x, y}$ is defined as:

$m \tuple {x, y} := \tuple {m x, m y}$


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 $\tuple {a, b} \in \C$, where:

$\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$
$\overline {\tuple {a, b} } = \tuple {\overline a, -b}$

where:

$\overline a$ is the conjugate of $a$

and

$\overline {\tuple {a, b} }$ is the conjugation operation on $\C$.

From Real Numbers form Algebra, $\overline a = a$ and so the above translate into:

$\tuple {a, b} \tuple {c, d} = \tuple {a c - d b, a d + c b}$
$\overline {\tuple {a, b} } = \tuple {a, -b}$


It is clear by direct comparison with the formal definition that this construction genuinely does generate the complex numbers.


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 $3 + 4 i$.


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 on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\map \Re z$ or $\mathop \Re z$.


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 on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\map \Im z$ or $\Im z$.


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 if and only if $b = 0$.


Wholly Imaginary

A complex number $z = a + i b$ is wholly imaginary if and only if $a = 0$.


Irrational

A complex number is classified as irrational if and only if its imaginary part is non-zero.


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


This representation is known as the complex plane.


Real Axis

Complex numbers of the form $\tuple {x, 0}$, being wholly real, appear as points on the $x$-axis.

Hence the $x$-axis of the complex plane is known as the real axis.


Imaginary Axis

Complex numbers of the form $\tuple {0, y}$, being wholly imaginary, appear as points on the points on the $y$-axis.

Hence the $y$-axis of the complex plane is known as the imaginary axis.


Polar Form

Polar-Form.png

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

\(\ds r\) \(=\) \(\ds \cmod z = \sqrt {x^2 + y^2}\) the modulus of $z$, and
\(\ds \theta\) \(=\) \(\ds \arg z\) the argument of $z$ (the angle which $z$ yields with the real line)

where $x, y \in \R$.

From the definition of $\arg z$:

$(1): \quad \dfrac x r = \cos \theta$
$(2): \quad \dfrac y r = \sin \theta$

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:

$z = x + i y = r \paren {\cos \theta + i \sin \theta}$

The pair $\polar {r, \theta}$ is called the polar form of the complex number $z \ne 0$.


The number $z = 0 + 0 i$ is defined as $\polar {0, 0}$.


Also known as

Some sources refer to a complex number as an imaginary number, but it is too easy to confuse this with a wholly imaginary number.

Hence this usage is not endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$.


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.


Historical Note

The concept of a complex number originated in the $16$th century as during the course of developing the solution to the general cubic equation.

In his Artis Magnae, Sive de Regulis Algebraicis of $1545$, Gerolamo Cardano considered the simultaneous equations:

$\begin {cases} x + y & = 10 \\ x y & = 40 \end {cases}$

and obtained the solution:

$\begin {cases} x & = 5 + \sqrt {-15} \\y & = 5 - \sqrt {-15} \end {cases}$

He made no attempt to interpret the meaning of the square root of a negative number, dismissing it with the comment:

So progresses arithmetic subtlely, the end result of which ... is as refined as it is useless.

On the other hand, he applied what is now known as Cardano's Formula to obtain a solution to:

$x^3 = 15 x + 4$

which leads to the expression:

$x = \sqrt [3] {2 + \sqrt {-121} } + \sqrt [3] {2 - \sqrt {-121} }$

whereas the "obvious" answer is $x = 4$.


Rafael Bombelli responded by treating $\sqrt {-121}$ in the same way as conventional numbers, showing that:

$\paren {2 \pm \sqrt {-1} }^3 = 2 \pm \sqrt {-121}$

from which we obtain:

$x = \paren {2 + \sqrt {-1} } + \paren {2 - \sqrt {-1} } = 4$


René Descartes, in his La Géométrie of $1637$, distinguished between "real numbers" and "imaginary numbers", concluding that if the latter occurred during the solution of a problem, it was in fact insoluble.

This view was endorsed by Isaac Newton.

However, by the $18$th century, complex numbers had gained acceptance.


Sources

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