Complex Number

⚫What is Complex number ?
⚪ Combination form of real and imaginary number
z = a + ib where i(iota) = √-1

⚫ Integer power of i (√-1)
(a) i⁴ⁿ = 1
(b) i⁴ⁿ⁺¹ = i
(c) i⁴ⁿ⁺² = -1
(d) i⁴ⁿ⁺³ = -i

⚫ Equality of two complex numbers
⚪ Two complex number z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ are said to be equal if and only if their real and imaginary parts are separately equal.
z₁ = z₂ ⇔ x₁ + iy₁ = x₂ + iy₂ ⇔ x₁ = x₂ and y₁ = y₂

⚫ Conjugate of a complex number :
⚪ For any complex number z = a + ib , then its conjugate is defined as z = a - ib
⚪ Geometrically , the conjugate of z is the reflection or point image of z in the real axis.

⚫ Properties of conjugate :
⚪ z̿ = z
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⚪ z + z̅ = 2Re(z) = purely real
⚪ z - z̅ = 2iIm(z) = purely imaginary
⚪ zz̅ = |z|² = purely real
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⚫ Modulus of a complex number
⚪ Modulus of a complex number z = x+ iy is defined as the distance of the point (x,y) from the origin.

⚫ Properties of modulus
⚪ |z| = |z| = |-z| = |-z| =|zi|
⚪ zz = |z|² = |z|²
⚪ |z₁z₂| = |z₁||z₂|
⚪ |z₁z₂z₃......z_n| = |z₁||z₂|.....|z_n|
⚪ |zⁿ| =|z|ⁿ
⚪ |z₁ ± z₂ |² = |z₁|² + |z₂|² ± (z₁z₂ + z₂z₁ ) = |z₁|² + |z₂|² ± 2Re(z₁z₂)
⚪ |z₁ + z₂ |² = |z₁|² + |z₂|² ⇒ z₁/z₂ is purely imaginary
⚪ |z₁ + z₂ |² + |z₁ - z₂ |² = 2 { |z₁|² + |z₂|² }

⚫ Argument of a complex number
⚪ Let z = a + ib be any complex number. If this complex number is represented geometricaaly by a point P , then the angle made by the OP with real axis is known as argument or amplitude of z.

⚫ De Moivre's Theorem
⚪ If n is any rational number , then ( cosθ + isinθ )ⁿ = cos nθ + isin nθ ⚪ If z =r(cosθ + i sinθ) and n is a positive integer , then

⚫ Triangle inequalities
⚪ In any triangle , sum of any two sides is greater than the third side and the difference of any two side is less than the third side.
⚪ |z₁ + z₂| ≤ |z₁| + |z₂|
⚪ |z₁ - z₂| ≤ |z₁| + |z₂|
⚪ |z₁ + z₂| ≥ | |z₁| - |z₂| |
⚪ |z₁ - z₂| ≥ | |z₁| - |z₂| |

⚫ Roots of a Complex Number
⚪ n^th roots of complex number
Let z = r (cos θ + i sin θ) be a complex number . By using De ' moivre 's theorem n^th roots having n distinct values of a complex numbers

⚪ The n^th roots of unity xⁿ = 1 = cos 0^o + i sin 0^o = cos 2kπ + i sin 2kπ
x = [cos 2kπ + i sin 2kπ]^1/n
Using De Moivre's Theorem

⚫ Properties of nth roots of unity
⚪ The sum of all n roots of unity is zero
1 + a + a² + ..... + a^n-1 = 0
⚪ Product of all n roots of unity is (-1)^n-1

⚫ Cube roots of unity
x³ - 1 = 0 ⇒ x = (1)^1/3

⚪ Properties of cube roots of unity
⚫ 1 + ω + ω² = 0
⚫ ω³ = 1
⚫ 1 + ωⁿ + ω²ⁿ = 0 { if n is not a multiple of 3 }
⚫ 1 + ωⁿ + ω²ⁿ = 3 { if n is a multiple of 3 }