maths > complex-number

Representation of Complex Numbers (incomplete)

what you'll learn...

Overview

Complex number Representation

»  Abstraction of complex numbers
→  solution of algebraic equations in 2D plane
→  eg: x3=1${x}^{3} = 1$ has 3$3$ solutions, (31)1st${\left(\sqrt[3]{1}\right)}_{1 s t}$, (31)2nd${\left(\sqrt[3]{1}\right)}_{2 n d}$, (31)3rd${\left(\sqrt[3]{1}\right)}_{3 r d}$

»  complex numbers are numerical expressions
→  eg: $3 + 4 \sqrt[4]{- 5}$ is one solution to ${\left(\frac{x - 3}{4}\right)}^{4} = - 5$

»  solution to quadratic equation in the form $a + i b$
→  $i = \sqrt{- 1}$

Note: irrational numbers do not have a generic form. But, it is later proven that all complex numbers can be expressed in the form $a + i b$.

For now, only solution to quadratic equation is shown to be in the form $a + i b$.

Representations of Complex Numbers

»  coordinate form $a + i b$

»  polar form $r \left(\cos \theta + i \sin \theta\right)$
abstracted from $a + i b$
→  $r = \sqrt{{a}^{2} + {b}^{2}}$
→  $\cos \theta = \frac{a}{r}$
→  $\sin \theta = \frac{b}{r}$

complex numbers

When using the real numbers, we come across problems that are mathematically modeled as ${x}^{2} = - 1$. There is no solution to this in real number system. So, the number system is extended beyond real numbers. This number system that is over-and-above the real number system and is named as 'complex numbers'.

Irrational numbers are represented with numerical expressions or symbols.
How a complex number is represented? In this topic an incomplete explanation to the form of representing complex number is discussed.

In the topic 'Generic form of Complex Numbers', the information discussed in here is further developed to complete the explanation. Understanding this complete representation with generic form requires some basics to be explained. In due course, the basics are explained and then we'll take up that.

In irrational number system, the solution to ${x}^{2} = 2$ is given as $\pm \sqrt{2}$.
Learning from that, the solution to ${x}^{2} = - 1$ is '$\pm \sqrt{- 1}$'. The solution is represented as a numerical expression (much the same way as irrational number representation).

Note: In the attempt to develop knowledge in stages, the discussion given in here are specific to quadratic equations. A more generic discussion will follow once first level of knowledge is acquired.

The solution to ${x}^{2} = - 4$ is '$\pm 2 \sqrt{- 1}$'.

The solution to ${\left(x + 3\right)}^{2} = - 4$ is '$- 3 \pm 2 \sqrt{- 1}$'

It is noted that a quadratic equation of the form $p {x}^{2} + q x + r = 0$ can be re-arranged to ${\left(x + \frac{q}{2 p}\right)}^{2} = - \frac{r}{p} + {\left(\frac{q}{2 p}\right)}^{2}$.
For any equation in this form, we can arrive at a solution in the form $a + b \sqrt{- 1}$.

$\sqrt{- 1}$ is represented with a letter $i$
The solution to a quadratic equation is in the form $a + b i$.

Note : The said explanation covers only solutions to quadratic equations. Let us examine this representation in detail and then later generalize this for complex numbers.

What are the solutions to the equation ${\left(x - 1\right)}^{2} = - 16$?
The answer is '$1 \pm 4 i$'.

summary

[Initial understanding] -- Form of complex number : Solution to quadratic equation is $a + b i$ where $a , b \in \mathbb{R}$ and $i = \sqrt{- 1}$

equivalent representation

All the following equals $2$
•  $4 - 2$
•  $1 + 1$
•  $\frac{6}{3}$

A number can be equivalently represented with numerical expressions of various forms.

Given hypotenuse is $c$. All the following represent the length of side opposite to the angle $\theta$ in a right angle triangle:
•  $c \sin \theta$
•  $c \tan \theta \cos \theta$
•  $c \sqrt{1 - {\cos}^{2} \theta}$

A constant can be represented as expressions.

It was established that solution to quadratic equation is in the form $a + b i$. An equivalent expression of this is

A number $a + b i$ is equivalently given as

This form looks similar to $\textcolor{b l u e}{\cos}$ and $\textcolor{g r e e n}{\sin}$ of a right angled triangle.

It is equivalently represented as $\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{r} \left(\textcolor{b l u e}{\cos \theta} + i \textcolor{g r e e n}{\sin \theta}\right)$ where $\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{r = \sqrt{{a}^{2} + {b}^{2}}}$ and $\theta = {\tan}^{- 1} \left(\frac{b}{a}\right)$.

We know that, $\sin \theta$ equals $\sin \left(\theta + 2 n \pi\right)$ where $n = 0 , 1 , 2 , \ldots$
$a + b i$

summary

A number in the form $a + b i$ is equivalently given as

where

$r = \sqrt{{a}^{2} + {b}^{2}} \theta = {\tan}^{- 1} \left(\frac{b}{a}\right)$

Outline