Complex Number

Complex number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number.

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Leonhard Euler in his fundamental work " Complete introduction to algebra
" (1770), noting the mysterious unreal nature of imaginary numbers, regarded them as a product of the imagination:“
Square roots of negative numbers are not equal to zero, are not less than zero, and are not greater than zero. From this it is clear that the square roots of negative numbers cannot be among the possible (actual, real) numbers. Hence, we have no another way except to acknowledge these numbers as impossible ones. This leads us to the notion of numbers, impossible in essence, which are usually called imaginary (fictitious) numbers, because they exist only in our imagination.”  

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Ref- http://shpenkov.janmax.com/ImaginUnitEng.pdf

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Complex Numbers in Real Life

There are two distinct areas that would want to address when discussing complex numbers in real life:

1. Real-life quantities that are naturally described by complex numbers rather than real numbers;
2. Real-life quantities which, though they're described by real numbers, are nevertheless best understood through the mathematics of complex numbers.

Example ->
Population A, 236 people, 48 of them children. Population B, 1234 people, 123 of them children. You might say that the fraction of children in population A is 48/236 while the fraction of children in population B is 123/1234, and that 48/236 (approx. 0.2) is much less than 123/1234 (approx. 0.1), so population A is a much younger population on the whole.

Now point out that you have used fractions, non-integer numbers, in a problem where they have no physical relevance. You can't measure populations in fractions; you can't have "half a person", for example. The kind of numbers that have direct relevance to measuring numbers of people are the natural numbers; fractions are just as alien to this context as the complex numbers are alien to most real-world measurements. And yet, despite this, allowing ourselves to move from the natural numbers to the larger set of rational numbers enabled us to deduce something about the real world situation, even though measurements in that particular real world situation only involve natural numbers.

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Ref -> https://www.math.toronto.edu/mathnet/questionCorner/complexinlife.html

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Definition
A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = −1. For example, −3.5 + 2i is a complex number.
The real number a is called the real part of the complex number a + bi; the real number b is called the imaginary part of a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part. The real part of a complex number z is denoted by Re(z) or ℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). 

 
Conjugate

The complex conjugate of the complex number z = x + yi is defined to be x − yi. It is denoted ${\displaystyle {\bar {z}}}$ or z*.

Addition and subtraction
Complex numbers are added by adding the real and imaginary parts of the summands. That is to say: ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i .

{(a+bi)+(c+di)=(a+c)+(b+d)}

Similarly, subtraction is defined by ( a + b i ) − ( c + d i ) = ( a − c ) + ( b − d ) i . {(a+bi)-(c+di)=(a-c)+(b-d)i }

Multiplication and division

In particular, the square of the imaginary unit is −1:

${\displaystyle i^{2}=i\times i=-1.\ }$

Reciprocal

Square root

 

Multiplication and division in polar form

 

we may derive

${\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).\,}$

Summary 

Equality of complex numbers

a+bi=c+di⟺a=c  and  b=d

Addition of complex numbers

(a+bi)+(c+di)=(a+c)+(b+d)i

Subtraction of complex numbers

(a+bi)−(c+di)=(a−c)+(b−d)i

Multiplication of complex numbers

(a+bi)⋅(c+di)=(ac−bd)+(ad+bc)i

Division of complex numbers 

a+bic+di=a+bic+di⋅c−dic−di=ac+bdc2+d2+bc−adc2+d2i

Polar form of complex numbers
a+bi=r⋅(cosθ+isinθ)

Related Topics ->

 Nyquist Plot

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