Complex Numbers in Number System

Overview

** Number System Hierarchy**

» Natural or counting numbers

→ Counting with hands 1, 2, 3, ...

» whole numbers

→ 0 is included

» integers

→ negatives are included.

» rational numbers

→ fractions and decimals are included.

» real numbers

→ irrational numbers included. *numbers represented as expressions*

» complex number : $a+ib$ form

→ $a$ is the real part

→ $b$ is the imaginary part

→ $i=\sqrt{-1}$

→ Complex means "consisting of many different parts"

**Irrational Number Representation**

» irrational numbers

→ solutions of algebraic equations in real line

→ eg: $\sqrt{2}$ is one solution of ${x}^{2}=2$

» irrational numbers are represented as *numerical expressions*

→ eg: $3+4\sqrt[3]{5}$ is one solution of ${\left(\frac{x-3}{4}\right)}^{3}=5$

hierarchy is arrangement

The whole numbers are '$0,1,2,3,...$'. This is used to count or measure quantities.

When using the whole numbers, we come across problems that are mathematically modeled to $x+1=0$. There is no solution to this in whole number system. So the number system is extended to integers or directed numbers. Whole numbers are extended to integers by including negative numbers.

In other words, whole numbers representation is not sufficient to represent directed numbers and so integers are defined.

For example, consider the numbers in

• I received $3$ candies and

• I gave $3$ candies.

In the whole numbers, both these are represented as $3$.

In integers, the first is $+3$ and the second is $-3$.*
Integer numbers are represented as follows.
$3$ is represented as either $\text{received:}3$ or $\text{aligned:}3$.
$-3$ is represented as either $\text{given:}3$ or $\text{opposed:}3$.*

The integers are '$...,-3,-2,-1,0,1,2,3,...$'.

When using the integers, we come across problems that are mathematically modeled to $2x=1$. There is no solution to this in integer number system. So the number system is extended to rational numbers which include fractions or decimals (which are different notation for same type of numbers).

In other words, whole numbers and integers representation is not sufficient to represent quantities like part of an object and so rational numbers are defined.

For example, A pizza is cut into $8$ pieces.

• $3$ whole pizzas and $5$ pieces of a cut pizza are remaining.

Whole numbers or integers represent them as two quantities:
$3$ pizzas and $5$ pieces when one whole is cut into $8$ pieces. This representation is descriptive.

The same in fractions and decimals is $3\frac{5}{8}=3.625$.

The rational numbers are 'numbers that can be represented as $\frac{p}{q}$ where $q\ne 0$.

When using the rational numbers, we come across problems that are mathematically modeled to ${x}^{2}=2$. There is no solution to this in rational number system. So the number system is extended to irrational numbers.

In other words, rational numbers are not sufficient to represent quantities or constants.

For example, a person has a square of $1m$ and want to represent the length of the diagonal. The diagonal is $\sqrt{2}m$.

The irrational numbers are 'numbers that cannot be represented as $\frac{p}{q}$ and are on the number line'

The combination of Rational and irrational numbers is 'real numbers'

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

Learning about the extended numbers is the objective of this topic "complex numbers".

summary

** Number System : **

•
Whole numbers

•
Integers (Whole numbers extended for negative numbers)

•
Rational Numbers (Integers extended for fractions)

•
Real numbers (Rational numbers extended for irrational numbers)

•
Complex numbers (Real numbers extended to include solutions to polynomials)

irrational number notation

Irrational numbers are 'numbers that cannot be represented as $\frac{p}{q}$ and are on the number line.

The equation ${x}^{2}=2$ has the solutions in irrational numbers. We try to write the solution in number form as
$\pm 1.4$
$\pm 1.41$
$\pm 1.414$
It is noted that none of the above is the exact solution and using the decimal representation, the number cannot be written down.

So, the solution is given as a numerical expression $\pm \sqrt{2}$. This is a numerical expression involving a square root symbol.

Consider ${(x-1)}^{2}=2$ The solution is $\pm \sqrt{2}+1$. Note, the solution is given as a numerical expression, involving a square root symbol and an addition.

The ratio of circumference to diameter of a circle is approximately $\frac{22}{7}$ or $3.14$. The exact value is represented with a symbol or a letter $\pi$.

Note: *The irrational number is represented with a letter*.

$e$ (a constant, that is the base of natural logarithm) is another irrational number.

summary

** Representation of Irrational Numbers : ** Irrational numbers are represented with numerical expressions (eg: $\sqrt[3]{4}+3$) or with symbols (eg: $\pi ,e$)

Outline

The outline of material to learn "complex numbers" is as follows.

Note : * Click here for detailed overview of Complex-Numbers *

→ __Complex Numbers in Number System__

→ __Representation of Complex Number (incomplete)__

→ __Euler's Formula__

→ __Generic Form of Complex Numbers__

→ __Argand Plane & Polar form__

→ __Complex Number Arithmetic Applications__

→ __Understanding Complex Artithmetics__

→ __Addition & Subtraction__

→ __Multiplication, Conjugate, & Division__

→ __Exponents & Roots__

→ __Properties of Addition__

→ __Properties of Multiplication__

→ __Properties of Conjugate__

→ __Algebraic Identities__