Definitions [3]
Definition: 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
Definition: Conjugate of a Complex Number
Conjugate of a complex number z = (a + ib) is defined as \[\overline{z}=\mathrm{a-ib}\].
Definition: Modulus of a Complex Number
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)
Key Points
Key Points: Properties of Conjugate of a Complex Number
- 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)
Key Points: Modulus of a Complex Number
- |z| = √(a² + b²)
- |z| = 0 ⇔ z = 0
- −|z| ≤ Re(z) ≤ |z|; −|z| ≤ Im(z) ≤ |z|
- |z₁z₂| = |z₁| |z₂|
- \[\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}\], z₂ ≠ 0
- |zⁿ| = |z|ⁿ
- |z₁ + z₂|² = |z₁|² + |z₂|² + 2Re(z₁ z̄₂)
- |z₁ − z₂|² = |z₁|² + |z₂|² − 2Re(z₁ z̄₂)
- |z₁ + z₂|² + |z₁ − z₂|² = 2(|z₁|² + |z₂|²)
- |z₁ + z₂| ≤ |z₁| + |z₂|
- |z₁ − z₂| ≥ ||z₁| − |z₂||
- z·z̄ = |z|²
- z₁z̄₂ + z̄₁z₂ = 2|z₁||z₂| cos(θ₁ − θ₂), where θ₁ = arg(z₁) and θ₂ = arg(z₂)
