Algebraic Operations on Complex Numbers 

1.0Introduction to Complex Numbers

A complex number is a number of the form z=a+ib, where a and b are real numbers, and $i=\sqrt{-1}$​.

• Real Part: a is the real part of z, denoted as Re(z).
• Imaginary Part: b is the imaginary part of z, denoted as Im(z). The set of all complex numbers is denoted by C. The introduction of complex numbers extends the real number system, allowing for solutions to equations like $x^2+1=0$

2.0Equality of Complex Numbers

Two complex numbers $z_1=a+ib,and{z}_2=c+id$ are equal if and only if their real parts are equal and their imaginary parts are equal.

$z_1=z_2\iff a=candb=d$

This property is crucial for solving equations involving complex numbers by equating the real and imaginary parts separately.

Example: If (x+2)+i(y−3)=5+2i, find the values of x and y. Equating the real parts: x+2=5⟹x=3. Equating the imaginary parts: y−3=2⟹y=5.

3.0Fundamental Algebraic Operations

The rules for working with complex numbers are the same as those for working with real numbers, with the important difference that you must always replace i² with -1.

Addition of Complex Numbers

If z₁ = a + ib and z₂ = c + id, then their sum is defined as: z₁ + z₂ = (a + c) + i(b + d)
In simple terms: Add the real parts together and add the imaginary parts together.

Example: Let z₁ = 3 + 5i and z₂ = -2 + 7i ⟹ z₁ + z₂ = (3 + (-2)) + i(5 + 7) = 1 + 12i.

Subtraction of Complex Numbers

If z₁ = a + ib and z₂ = c + id, then their difference is defined as: z₁ - z₂ = (a - c) + i(b - d)
In simple terms: Subtract the real parts and subtract the imaginary parts.

Example: Let z₁ = 3 + 5i and z₂ = -2 + 7i. ⟹ z₁ - z₂ = (3 - (-2)) + i(5 - 7) = 5 - 2i.

Multiplication of Complex Numbers

If z₁ = a + ib and z₂ = c + id, then their product is defined by multiplying them binomially and using i² = -1: z₁ * z₂ = (a + ib)(c + id) = ac + iad + ibc + i²bd = (ac - bd) + i(ad + bc)

Example: Let z₁ = 3 + 5i and z₂ = -2 + 7i.
z₁ * z₂ = (3)(-2) + (3)(7i) + (5i)(-2) + (5i)(7i)
= -6 + 21i -10i + 35i²
= -6 + 11i + 35(-1)  ...since i² = -1
= -6 + 11i - 35 = -41 + 11i.

Division of Complex Numbers

Division is the most intricate operation. The goal is to write the result back in the standard a + ib form. If z₁ = a + ib and z₂ = c + id ≠ 0, then: z₁ / z₂ = (a + ib) / (c + id)

The process involves multiplying both the numerator and the denominator by the complex conjugate of the denominator (see Section 5). This eliminates the imaginary part in the denominator.
z₁ / z₂ = [(a + ib) / (c + id)] * [(c - id) / (c - id)] = [(ac + bd) + i(bc - ad)] / (c² + d²)

Example: Let z₁ = 3 + 5i and z₂ = 1 - 2i.
z₁ / z₂ = (3 + 5i) / (1 - 2i)
Multiply numerator and denominator by the conjugate of the denominator, (1 + 2i):
= [(3 + 5i)(1 + 2i)] / [(1 - 2i)(1 + 2i)]
Numerator: (3 + 5i)(1 + 2i) = 3 + 6i + 5i + 10i² = 3 + 11i -10 = -7 + 11i
Denominator: (1 - 2i)(1 + 2i) = 1² - (2i)² = 1 - 4i² = 1 - 4(-1) = 5
Therefore, z₁ / z₂ = (-7 + 11i) / 5 = -7/5 + (11/5)i.

4.0Properties of Algebraic Operations

For any complex numbers z, z₁, z₂, z₃:

1. Closure Law: z₁ + z₂ ∈ C and z₁ * z₂ ∈ C. (The sum and product are also complex numbers).
2. Commutative Law: z₁ + z₂ = z₂ + z₁ and z₁ * z₂ = z₂ * z₁.
3. Associative Law: (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃) and (z₁ * z₂) * z₃ = z₁ * (z₂ * z₃).
4. Distributive Law: z₁ * (z₂ + z₃) = z₁ * z₂ + z₁ * z₃.
5. Additive Identity: The number 0 = 0 + i0 is such that z + 0 = z for every z ∈ C.
6. Multiplicative Identity: The number 1 = 1 + i0 is such that z * 1 = z for every z ∈ C.
7. Additive Inverse: For z = a + ib, its additive inverse is -z = -a - ib.
8. Multiplicative Inverse: For a non-zero complex number z = a + ib, its multiplicative inverse (or reciprocal) z⁻¹ is a number such that z * z⁻¹ = 1.
   z⁻¹ = 1/(a+ib) = (a - ib)/(a² + b²) = $\bar{z}/|z{|}^2$

5.0The Conjugate of a Complex Number

The conjugate of a complex number z = a + ib is denoted by $\bar{z}$ and is defined as $\bar{z}=a-ib$. Geometrically, it represents the mirror image of the point (a, b) about the real (x-) axis.

Properties of Conjugates:

• $\overline{(z)}$ = z
• $z+\bar{z}=2Re(z)$
• $z-\bar{z}=2iIm(z)$
• $z\bar{z}=|z{|}^2=a^2+b^2$ (A very important property used in division)
• $\overline{(z_1+z_2)}={\bar{z}}_1+{\bar{z}}_2$
• $\overline{(z_1-z_2)}={\bar{z}}_1-{\bar{z}}_2$
• $\overline{(z_1\ast z_2)}={\bar{z}}_1\ast {\bar{z}}_2$
• $\overline{(z_1/z_2)}={\bar{z}}_1/{\bar{z}}_2,$ provided z₂ ≠ 0

6.0The Modulus and Argument

The modulus (or absolute value) of a complex number z = a + ib is its distance from the origin in the Argand plane. It is denoted by |z| and is a non-negative real number.
|z| = √(a² + b²)

The argument (or amplitude) of a complex number z = a + ib (≠ 0) is the angle θ that the line joining the point to the origin makes with the positive real axis. It is denoted by arg(z) or amp(z).
arg(z) = θ = tan⁻¹(b/a), where the quadrant of the point (a, b) must be considered.

Properties of Modulus:

• |z| ≥ 0 and |z| = 0 iff z = 0
• |z| =$|\bar{z}|$ = |-z|
• |z₁ * z₂| = |z₁| * |z₂|
• |z₁ / z₂| = |z₁| / |z₂|, provided z₂ ≠ 0
• |z₁ + z₂| ≤ |z₁| + |z₂| (Triangle Inequality - Very important for JEE)
• |z₁ - z₂| ≥ ||z₁| - |z₂||