Revision: 11th Std >> Complex Numbers MAH-MHT CET (PCM/PCB) Complex Numbers

z = x + iy, x, y∈ R and \[i=\sqrt{-1}\] is called a complex number. 

x ⇒ Real Part Re(z)

iy ⇒ Imaginary Part Im(z)

If Re(z) = x = 0, then the complex number z is purely imaginary.

If Im(z) =y = 0, then complex number z is purely real.

Integral powers of iota (i):

\[\mathrm{i}^2=-1\]

\[\mathrm{i}^3=-\mathrm{i}\]

\[\mathrm{i}^{4}=1\]

In general,

\[1^{4n}=1\], \[\mathrm{i^{4n+1}=i}\], \[\mathrm{i^{4n+2}=-1}\], \[\mathrm{i^{4n+3}=-i}\] ...where n ∈ N

Two complex numbers z₁ = a + ib and z₂ = c + id are said to be equal, if

Re(z₁) = Re(z₂) ⇒ a = c

Im(z₁) = Im(z₂) ⇒ b = d

Conjugate of a complex number z = (a + ib) is defined as \[\overline{z}=\mathrm{a-ib}\].

∴ z = a + ib = r(cos θ + i sin θ) = re^iθ,

where r = |z| and θ = arg z is called an exponential form of a complex number.

The polar form of a complex number z = x + iy is

z = r(cos θ + i sin θ), where x = r cos θ, y = r sin θ and r = \[r=\sqrt{x^{2}+y^{2}}\].

The modulus (or absolute value) of a complex number, z = a + ib, is defined as the non-negative real number

√(a² + b²). It is denoted by |z| i.e. |z| = √(a² + b²)

If a complex number is given by z = x + iy, then the argument of z i.e. θ is defined as \[\theta=\arg\left(z\right)=\tan^{-1}\left(\frac{y}{x}\right),0\leq\theta<2\pi\]

The cube roots of unity are the solutions of the equation
x³ = 1

They are: 1, \[\frac{-1+i\sqrt{3}}{2}\], \[\frac{-1-i\sqrt{3}}{2}\]

They are denoted by 1, ω, ω²

Negative Powers

(cos θ + i sin θ)⁻ⁿ = cos(nθ) − i sin(nθ)

Conjugate Form

(cos θ − i sin θ)ⁿ = cos(nθ) − i sin(nθ)

Reciprocal Form

\[\frac{1}{\left(\cos\theta+\mathrm{i}\sin\theta\right)^1}=\cos\theta-\mathrm{i}\sin\theta\]

\[(\cos\theta-\mathrm{i}\sin\theta)^{-\mathrm{n}}=\cos\mathrm{n}\theta+\mathrm{i}\sin\mathrm{n}\theta\]

1. Quadratic Equation
   ax² + bx + c = 0, where a ≠ 0
2. Roots Formula
   \[x=\frac{-\mathrm{b}+\sqrt{\mathrm{b}^{2}-4\mathrm{ac}}}{2\mathrm{a}}\]

Discriminant

D = b² − 4ac

• If D < 0 → roots are complex
• If D = 0 → roots are real and equal

Conjugate Roots

1. If p + iq is a root, then p − iq is also a root
   ⇒ Complex roots occur in conjugate pairs

(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

• Double Conjugate
  z̄̄ = z
• Sum with Conjugate
  z + z̄ = 2 Re(z)
• Difference with Conjugate
  z − z̄ = 2i Im(z)
• Purely Real Condition
  z = z̄ ⇔ z is purely real
• Purely Imaginary Condition
  z + z̄ = 0 ⇔ z is purely imaginary
• Addition
  \[\overline{z_{1}+z_{2}}=\overline{z}_{1}+\overline{z}_{2}\]
• Subtraction
  \[\overline{z_1-z_2}=\overline{z}_1-\overline{z}_2\]
• Multiplication
  \[\overline{z_1z_2}=\overline{z}_1\overline{z}_2\]
• Division
  \[\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline{z}_1}{\overline{z}_2},z_2\neq0\]
• z · z̄ = [Re(z)]² + [Im(z)]²
• \[\overline{z^{n}}=\left(\overline{z}\right)^{n}\]
• z₁z̄₂ + z̄₁z₂ = 2 Re(z₁z̄₂)

Let √(a + ib) = x + iy

1. Square both sides
   (x + iy)² = a + ib
2. Expand
   x² − y² + 2ixy = a + ib
3. Equate real and imaginary parts
   x² − y² = a
   2xy = b
4. Solve these equations to find x and y
5. Then, √(a + ib) = ±(x + iy)

1. Representation

• z = a + ib → point (a, b)
• X-axis → Real part (Re)
• Y-axis → Imaginary part (Im)

2. Modulus

• |z| = distance from origin
• |z| = √(a² + b²)

3. Argument (θ)

• Angle made with +X-axis (anticlockwise)
• θ = tan⁻¹(b/a)

1. |z| = √(a² + b²)
2. |z| = 0 ⇔ z = 0
3. −|z| ≤ Re(z) ≤ |z|; −|z| ≤ Im(z) ≤ |z|
4. |z₁z₂| = |z₁| |z₂|
5. \[\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}\], z₂ ≠ 0
6. |zⁿ| = |z|ⁿ
7. |z₁ + z₂|² = |z₁|² + |z₂|² + 2Re(z₁ z̄₂)
8. |z₁ − z₂|² = |z₁|² + |z₂|² − 2Re(z₁ z̄₂)
9. |z₁ + z₂|² + |z₁ − z₂|² = 2(|z₁|² + |z₂|²)
10. |z₁ + z₂| ≤ |z₁| + |z₂|
11. |z₁ − z₂| ≥ ||z₁| − |z₂||
12. z·z̄ = |z|²
13. z₁z̄₂ + z̄₁z₂ = 2|z₁||z₂| cos(θ₁ − θ₂), where θ₁ = arg(z₁) and θ₂ = arg(z₂)

• arg (any +ve real no.) = 0, arg (any +ve imaginary no.) = π/2

• arg (any −ve real no.) = π, arg (any −ve imaginary no.) = −π/2

• arg (z₁z₂) = arg(z₁) + arg(z₂)

• arg (z₁ / z₂) = arg(z₁) − arg(z₂)

• arg (z̄) = − arg(z) = arg (1/z)

• arg (+z) = π ± arg(z) and arg (−z) = arg(z) ± π
• arg (z) + arg (z̄) = 0

• arg(z₁z₂z₃…zₙ) = arg(z₁) + arg(z₂) + … + arg(zₙ)

• arg(zⁿ) = n arg(z)

• arg(z̄) = −arg(z)

• arg(z²) = 2 arg(z)

• If arg(z) = 0 ⇒ z is real

• ω³ = 1
• 1 + ω + ω² = 0
• ω² = 1/ω
• ω̄ = ω² and \[\left(\overline{\omega}\right)^2=\omega\]
• ω³ⁿ = 1
  ω³ⁿ⁺¹ = ω
  ω³ⁿ⁺² = ω²
• ω + ω² = −1
• ωω² = 1
• arg(ω) = \[\frac{2\pi}{3}\]
  arg(ω²) = \[\frac{4\pi}{3}\]

Set of Points in Complex Plane:

Let z = x + iy (variable point) and z₁ = x₁ + iy₁ (fixed point)