Introduction
Imaginary Numbers
Definition In Mathematics, the imaginary number i is defined such that
$\mathrm{i}^2 = -1$
. So,$\mathrm{i} = \sqrt{-1}$
.
Cartesian form of a complex number
$z = a + b\mathrm{i}$
, where $a,b \in \mathbb{R}$
is called the Cartesian form of a complex number.
$a$
is called the real part of $z$
, written as Re(z) = a
and $b$
is called the imaginary part of $z$
, written as Im(z) = b.
Geometric representation
Argand diagram
The real number line is extended to a plane of points called an Argand diagram. The plane is denoted by $\mathbb{C}$
.
Modulus
Given the complex number
$z = x + y\mathrm{i}$
,$x,y \in \mathbb{R}$
, the modulus of$z$
is given by$|z| = |x + y\mathrm{i}| = \sqrt{ x^2 + y^2} = \sqrt{\left(\mathrm{Re}(z)\right)^2 + \left(\mathrm{Im}(z)\right)^2}$
The modulus is the distance r from the origin.
Operations
Two complex numbers are equal …
addition
$z_1 + z _2 = (a + b\mathrm{i}) + (c + d\mathrm{i}) = (a + c) + (b + d)\mathrm{i}$
scalar multiplication
$\lambda z = \lambda (a + b\mathrm{i}) = (\lambda a) + (\lambda b)\mathrm{i}$
multiplication
$z_1 \cdot z _2 = (a + b\mathrm{i}) \cdot (c + d\mathrm{i}) = (ac - bd) + (ad + bc)\mathrm{i}$
conjugates
For every complex number $z = a + b\mathrm{i}$
there is a conjugate complex number of the form $z^* = a - b\mathrm{i}$
.
Their real parts are equal and their imaginary parts are opposite.
The conjugates are useful when dividing two complex numbers.
example
Find $\frac{1 + 3\mathrm{i}}{2 - \mathrm{i}}$
$\frac{1 + 3\mathrm{i}}{2 - \mathrm{i}} \cdot \frac{2 + \mathrm{i}}{2 + \mathrm{i}} = \frac{2 - 3 + 6\mathrm{i} + \mathrm{i}}{4 + 1} = \frac{1}{5}(-1 +7\mathrm{i})$
Powers and Roots
argument
The argument $\theta = \arg(z)$
is the angle formed by the complex number $z=a+ib$
in the Complex plane in anti-clockwise direction.
$$\theta = \arg(z) = \arctan{\left(\frac{b}{a}\right)}$$
Polar form
In the the Argand plane
$$x = r\cos{\theta}$$
$$y = r\sin{\theta}$$
where $r = |z|$
So the complex number $z = x+iy$
in Cartesian form can be written as
$$z= r(\cos{\theta} + i\sin{\theta})$$
short form
Some literature uses a short form:
$$z= r \cdot \mathrm{cis~} {\theta}$$
Euler’s or exponential form of a complex number
A complex number can written in Euler’s form.
$$z = r\mathrm{e}^{i\theta}$$
where $r = |z|$
and $\theta = \arg(z) = \arctan{\left(\frac{b}{a}\right)}$
opposite
conjugate
opposite-conjugate
operations with complex numbers in polar form
addition
multiplication by a scalar
multiplication of complex numbers in polar form
multiplying the moduli and adding the arguments
$$z_1 \cdot z_2$$
reciprocal complex number
$$\frac{1}{z}$$
- reciprocal modulus
- opposite argument
division of complex numbers in polar form
$$\frac{z_1}{z_2}$$
Powers and Roots of Complex Numbers in Polar Form
De Moivre’s theorem
$$z^n = r^n(\cos{n\theta} + i \sin{n\theta})$$
Also, it can be shown that $$z^{\frac{1}{n}} = r^{\frac{1}{n}}(\cos{\frac{\theta}{n}} + i \sin{\frac{\theta}{n}})$$
The periodic nature of the trigonometric functions allows for multiple roots.
Given
$$z^n = r(\cos{\theta} + i \sin{\theta})$$
Say, w is the nth root of z, where $z = r\mathrm{cis~}\theta$
.
Taking the nth root allows for multiples of $2\pi$
as $z = r\mathrm{cis~}(\theta + 2\pi k)$
all represent the same complex number.
So, $$w = z^{\frac{1}{n}} = r\left(\cos{\left(\frac{\theta+ 2\pi k}{n}\right)} + i \sin{\left(\frac{\theta+ 2\pi k}{n}\right)}\right)$$