Complex Number
A combination of real and imaginary numbers in the form of z = x + iy, where x, y ∈ R is called a complex number.
Real and Imaginary Parts of a Complex Number
Let z = x + iy be a complex number
- then ‘x’ is called the real part of z; Re (Z) and
- ‘y’ is called the imaginary part of z and it may be denoted as Im (Z)
Iota
Mathematician Euler introduced the symbol i (read as iota) for √-1 with the property i2 + 1 = 0 i.e. i2 = -1. He also called this symbol the imaginary unit.
- i = √-1
- i2 = -1
- i3 = – i
- i4 = 1
Algebra of Complex Numbers
- Addition of Complex Numbers
- Subtraction of Complex Numbers
- Multiplication of Complex Numbers
- Division of Complex Numbers
Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, then their sum will be defined as
(1) Addition of Complex Numbers
z1 + z2 = (x1 + iy1) + (x2 + iy2)
or, z1 + z2 = (x1 + x2) + i(y1 + y2)
Properties
- Closure Property – The sum of two complex numbers is also a complex number.
- Commutative Property – z1 + z2 = z2 + z1
- Associative Property – (z1 + z2)+ z3 = z1 + (z2 + z3)
(2) Subtraction of Complex Numbers
z1 – z2 = (x1 + iy1) – (x2 + iy2)
or, z1 + z2 = (x1 – x2) + i(y1 – y2)
Note: z1 – z2 follows the closure property, but this operation is neither commutative nor associative.
(3) Multiplication of Complex Numbers
z1 z2 = (x1 + iy1) × (x2 + iy2)
= (x1x2 – y1 y2) + i(x1y2 + x2y1)
Properties
- Closure Property – The sum of two complex numbers is also a complex number.
- Commutative Property – z1 × z2 = z2 × z1
- Associative Property – (z1 × z2)× z3 = z1 × (z2 × z3)
- Distributive Property – z1 × (z2 + z3) = z1 × z2 + z1 × z3
(4) Division of Complex Numbers
{z_1\over z_2} = {(x_1+iy_1)\over(x_2+iy_2)}Note: division of two complex numbers {z_1\over z_2} follows the closure property, but this operation is neither commutative nor associative.
Identities Related to Complex Numbers
For any complex number z1, z2, we have
- (z1 + z2)2 = z12 + 2z1z2 + z22
- (z1 – z2)2 = z12 – 2z1z2 + z22
- (z1 + z2)3 = z13 + 3z12z2 + 3z1z22+ z23
- (z1 – z2)3 = z13 – 3z12z2 + 3z1z22– z23
- z12 – z22 = (z1 + z2)(z1 – z2)
Conjugate of a Complex Number
Let z = x + iy is a complex number, then the conjugate of z is denoted by \overline{z}
i.e. \overline{z} = x – iy
Properties of Conjugate of Complex Number
For any complex number z, z1, z2, we have
- \overline {\overline{z} }=z
- \overline{z_1 + z_2} =\overline{z_1} + \overline{z_2}
- \overline{z_1 - z_2} =\overline{z_1} - \overline{z_2}
- \overline{z_1 . z_2} =\overline{z_1} . \overline{z_2}
- (\overline{{z_1 \over z_2}}) = {\overline{z_1}\over \overline{z_2} }
Reciprocal/Multiplicative Inverse of a Complex Number
Let z = x + iy is a non-zero complex number, then the multiplicative inverse
z-1 = 1\over z = 1\over x+iy
= {1\over x+iy} × {x-iy\over x-iy}
Modulus of a Complex Number
Let z = x + iy, then the modulus or magnitude of z is denoted by |z| and given by
|z| = \sqrt{x^2+y^2}Properties of Modulus
- |z| ≥ 0
- |z| = |\overline{z} |
- z\overline{z} = |z|2
- |z1| = |z2|
- |{z_1\over z_2}|={|z_1|\over|z_2|}
- |z1 ± z2| ≤ |z1| + |z2| and |z1 ± z2| ≥ |z1| – |z2|
- |zn| = |z|n
Argand Plane and Argument
Argand Plane: Any complex number z = x + iy can be represented geometrically by a point (x, y) in a plane, called the Argand plane or Gaussian plane.
The length of the line segment OZ is the modulus of z,
i.e. |z| = length of OZ = \sqrt{x^2+y^2}
Argument
The angle made by the line joining point z to the origin, with the positive direction of real axis is called argument of that complex number. It is denoted by the symbol arg (z) or amp (z).
arg (z) = θ = tan^{-1}({y\over x})
Principal Value of Argument
The value of the argument which lies in the interval (-π, π} is called the principal value of the argument.
- If x > 0 and y > 0, then arg (z) = θ
- If x < 0 and y > 0, then arg (z) = π – θ
- If x < 0 and y < 0, then arg (z) = – (π – θ)
- If x > 0 and y < 0, then arg (z) = – θ
where, θ = tan^{-1}({y\over x})
Polar Form of a Complex Number
If z = x + iy is a complex number, then z can be written as z = r(cos θ + i sin θ), where θ = arg (z) and r = \sqrt{x^2+y^2} this is called polar form.
Eulerian Form of a Complex Number
If z = x + iy is a complex number, then it can be written as
z = reiθ
Where, θ = arg (z) and r = \sqrt{x^2+y^2}
This is the Eulerian form and eiθ = cos θ + isin θ and e-iθ = cos θ – isin θ.
Cube root of Unity
Cube root of unity are 1, ω, ω2
Where, ω = -1+i \sqrt{3}\over 2
and -1-i \sqrt{3}\over 2
Properties of Cube root of unity
- 1 + ω + ω2 = 0
- ω3 = 1
Square Root of a Complex Number
If z = x + iy, then
\sqrt{z}=\sqrt{x+iy}\pm \Bigg[{ \sqrt{|z|+x} \over 2}+i{ \sqrt{|z|-x} \over 2}\Bigg], for y > 0 \pm \Bigg[{ \sqrt{|z|+x} \over 2}-i{ \sqrt{|z|-x} \over 2}\Bigg], for y < 0
