Complex Number Notes

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*Sample Notes of complex Numbers

Introduction:- Euler was the first mathematician to introduce the symbol i (iota) for the square root of  -1 with the property of  i² = -1. He also called this symbol as the imaginary unit.

Complex Number:-

            Any number of the form  z = x + iy is called complex number. (where x, y ɛ R).    

                                    X à Real part of Z à Re(Z)

                                    Y à Imaginary part of Z à Im(Z).

Iota (i):-

                        i² = -1       

                        i³ = i² . i à  -i

                        i⁴ = i² . i² à 1

                        i⁵ = i . i⁴ à i

** Sum of any 4 consecutive powers of i will be zero. Means (i + i² + i³ + i⁴ = 0).

Eq: Find the value of

                 i¹⁰ + i¹¹ + i¹² + ----------- + i³⁹

Comparison of complex numbers:-

            If Z₁ = x₁ +iy₁ and Z₂ = x₂ + iy₂ are two complex numbers then there can not be any comparison between Z₁ and Z₂ (Like Z₁ > Z₂ or Z₁ < Z₂).

                        But two complex numbers can be equal to each other if Z₁ = Z₁ then

_i.                     x₁ = x₂

_ii.                    y₁ = y₂    

Operations on complex numbers:-

If Z₁ = x₁ +iy₁ and Z₂ = x₂ + iy₂ are two complex numbers then-

i.                    Z₁ + Z₂ = (x₁ + x₂) + i(y₁ + y₂)

ii.                 Z₁ – Z₂ = (x₁ – x₂) + i(y₁ – y₂)

iii.               Z₁ . Z₂ = (x₁x₂ – y₁y₂) + i(x₁y₂ + x₂y₁)

iv.               Z₁/Z₂ = (x₁ + iy₁)/(x₂ + iy₂)

** All these operations are used everywhere in complex number and direct questions are like to convert complex number in the form of (x + iy).

Conjugate of a complex number:-

            Let Z = x + iy be a complex number, Then the conjugate of Z is denoted by Z and is equal to Z = x – iy.

** Conjugate of any complex number is obtain by replacing i by –i.

Properties of conjugate 

details are given in notes

 

Reciprocal of a complex number:-

Let Z = x + iy be a non-zero complex number, then

                SEE NOTES AND SOLVE 

**Do it by yourself

Ex:  Z = 3+2i  , Z = 4-5i

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