COMPLEX NUMBER
COMPLEX NUMBER
✤ Standard form. z = a+ib {I = iota , b = imaginary part}
↓
Real Part
Ex :- z=3+4i
Here a=3, b=4.
r = |z| (modulus of complex No.
= √a²+b²
Properties of i
1. i² = -1
2. i³ = -i
3. i⁴ = 1
§. 1. i⁴ᵐ = 1
2. i⁴ᵐ⁺ⁿ = iⁿ.
3. 1/i = -i
Note! when we have to solve higher power of i, we should divide the
power of i by 4. and use remainder.
Ex:- i²⁰²⁴ = i⁴ˣ⁵⁰⁶⁺⁰ = i⁰ = 1
§ Conjugate of Complex No is Represented by Z
If z = a+ib
then z = a-ib (we have to change the sign of i)
Prop. of conjugate
1. |z ̅| = |z ̅|
2. zz ̅ = |z ̅|²
3. z₁+z₂ = z ̅₁±z ̅₂
4. (z ̅₁/z₂) = z ̅₁/z ̅₂
❈ Argument: angle made by complex number with in argant Plane.
z = a+ib
Steps to find argument:
1) find tanθ = |b/a| and find θ.
(i) If (a,b) ∈ 1st quadrant then argument (α) = θ.
2) If (a,b) ∈ 2nd then α = π-θ
3) If (a,b) ∈ 3rd then α = -(π-θ)
4) If (a,b) ∈ 4yth then α = 2π-θ = -θ.
Ex z = -1 + i√3.
Here a=-1, b=√3
tanθ = |b/a|
= |√3/-1|
tanθ = √3
θ = π/3
Here
(a,b) ∈ 2nd then.
α = π - θ
α = π - π/3 = 2π/3
❈ Polar form of Complex No:
let z = a+ib is any Complex Number then sets to find Polar
form
1 find r = √a²+b².
2 find argument α
3 write as r(cosα+isinα)
Ex z = -1+ i√3
Here r = √(-1)²+(√3)²
= √1+3 = 2.
asin above Example α=2π/3
So Polar form
= 2(cos2π/3 + isin2π/3)
✤ Solution of Quadratic Equation.
let ax²+bx+c=0
If b²-4ac < 0
roots are complex in number
i.e in the terms of i
x = -b ± √b²-4ac/2a
in square root we use i²
in the place of -1
Ex: 1.x²-4x+13 = 0
Here a=1, b=-4, c=13
x = 4 ± √(16-4x1x13)/ 2x1
= 4 ± √(16-52)/2
= 4 ± √-36/2
= 4 ± √(-1x36)/2
x = 4 ± 6i/ 2
x= 2 ± 3i
Cube root of unity : It is represented by 'w'
1) w = -1+i√3/2 , w² = -1-i√3/2
2) 1+w+w²=0
3) w³=1
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