Complex Number

Section 1: Fundamentals of Complex Numbers

1.1 Introduction

The equation $x^2 + 1 = 0$ has no solution in the set of real numbers. To solve such equations, mathematicians introduced the concept of imaginary numbers.

1.2 Imaginary Unit ($i$)

The imaginary unit is denoted by $i$ and is defined as:

$$i = \sqrt{-1}$$

Properties of $i$:

• $i^2 = -1$
• $i^3 = i^2 \cdot i = -i$
• $i^4 = (i^2)^2 = 1$
• $i^{4n} = 1$, $i^{4n+1} = i$, $i^{4n+2} = -1$, $i^{4n+3} = -i$ (where $n$ is any integer)

Important Note:

For any positive real number $a$, $\sqrt{-a} = i\sqrt{a}$

1.3 Definition of Complex Number

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

• $a$ is called the real part of $z$, denoted as $Re(z) = a$
• $b$ is called the imaginary part of $z$, denoted as $Im(z) = b$

Standard Form:

$$z = a + ib, \quad a, b \in \mathbb{R}$$

1.4 Algebra of Complex Numbers

(i) Equality of Complex Numbers

Two complex numbers $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ are equal iff:

$$a_1 = a_2 \quad \text{and} \quad b_1 = b_2$$

(ii) Addition

If $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$, then:

$$z_1 + z_2 = (a_1 + a_2) + i(b_1 + b_2)$$

(iii) Subtraction

$$z_1 - z_2 = (a_1 - a_2) + i(b_1 - b_2)$$

(iv) Multiplication

$$z_1 \cdot z_2 = (a_1a_2 - b_1b_2) + i(a_1b_2 + a_2b_1)$$

(v) Division (for $z_2 \neq 0$)

$$\frac{z_1}{z_2} = \frac{a_1 + ib_1}{a_2 + ib_2} \times \frac{a_2 - ib_2}{a_2 - ib_2} = \frac{a_1a_2 + b_1b_2}{a_2^2 + b_2^2} + i\frac{a_2b_1 - a_1b_2}{a_2^2 + b_2^2}$$

1.5 Properties of Algebraic Operations

Properties of Addition:

1. Closure Law: For any two complex numbers $z_1, z_2$, $z_1 + z_2$ is also a complex number.
2. Commutative Law: $z_1 + z_2 = z_2 + z_1$
3. Associative Law: $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$
4. Additive Identity: There exists a complex number $0 + i0$ such that $z + 0 = z$
5. Additive Inverse: For every complex number $z = a + ib$, there exists $-z = -a - ib$ such that $z + (-z) = 0$

Properties of Multiplication:

1. Closure Law: $z_1 \cdot z_2$ is a complex number.
2. Commutative Law: $z_1 \cdot z_2 = z_2 \cdot z_1$
3. Associative Law: $(z_1 \cdot z_2) \cdot z_3 = z_1 \cdot (z_2 \cdot z_3)$
4. Multiplicative Identity: There exists a complex number $1 + i0$ such that $z \cdot 1 = z$
5. Multiplicative Inverse: For every non-zero complex number $z$, there exists $z^{-1}$ such that $z \cdot z^{-1} = 1$

Distributive Law:

$$z_1 \cdot (z_2 + z_3) = z_1z_2 + z_1z_3$$

1.6 Geometrical Representation of Complex Numbers

Argand Plane (Complex Plane)

A complex number $z = a + ib$ can be represented as a point $P(a, b)$ in the Cartesian plane.

• X-axis: Real axis ($a$)
• Y-axis: Imaginary axis ($b$)

Polar Representation

For a complex number $z = a + ib$:

$$z = r(\cos\theta + i\sin\theta)$$

where:

• $r = |z| = \sqrt{a^2 + b^2}$ (modulus)
• $\theta = \tan^{-1}(\frac{b}{a})$ (argument/amplitude)

Euler's Form:

$$z = re^{i\theta} = r(\cos\theta + i\sin\theta)$$

Section 2: Conjugate and Absolute Value

2.1 Conjugate of a Complex Number

For a complex number $z = a + ib$, its conjugate is denoted by $\bar{z}$ and is defined as:

$$\bar{z} = a - ib$$

Geometrical Meaning:

The conjugate $\bar{z}$ is the mirror image of $z$ in the real axis (X-axis).

2.2 Properties of Conjugate

For any complex numbers $z, z_1, z_2$:

(i) Conjugate of a Sum

$$\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$$

(ii) Conjugate of a Difference

$$\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}$$

(iii) Conjugate of a Product

$$\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}$$

(iv) Conjugate of a Quotient

$$\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}, \quad z_2 \neq 0$$

(v) Double Conjugate

$$\overline{\overline{z}} = z$$

(vi) Sum of a Number and its Conjugate

$$z + \overline{z} = 2a = 2Re(z)$$

(vii) Difference of a Number and its Conjugate

$$z - \overline{z} = 2ib = 2iIm(z)$$

(viii) Product of a Number and its Conjugate

$$z \cdot \overline{z} = a^2 + b^2 = |z|^2$$

(ix) Conjugate and Real/Imaginary Parts

• If $z = \overline{z}$, then $z$ is purely real $(b = 0)$
• If $z = -\overline{z}$, then $z$ is purely imaginary $(a = 0)$

2.3 Absolute Value (Modulus) of a Complex Number

For a complex number $z = a + ib$, the absolute value or modulus is denoted by $|z|$ and is defined as:

$$|z| = \sqrt{a^2 + b^2}$$

Geometrical Meaning:

$|z|$ represents the distance from the origin to the point $(a, b)$ in the Argand plane.

2.4 Properties of Absolute Value

(i) Non-negativity

$$|z| \geq 0, \quad |z| = 0 \iff z = 0$$

(ii) Product of Moduli

$$|z_1 \cdot z_2| = |z_1| \cdot |z_2|$$

(iii) Quotient of Moduli

$$\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}, \quad z_2 \neq 0$$

(iv) Modulus of Conjugate

$$|\overline{z}| = |z|$$

(v) Modulus of Power

$$|z^n| = |z|^n, \quad n \in \mathbb{N}$$

(vi) Triangle Inequality

$$|z_1 + z_2| \leq |z_1| + |z_2|$$

(vii) Reverse Triangle Inequality

$$|z_1 - z_2| \geq ||z_1| - |z_2||$$

(viii) Modulus and Real/Imaginary Parts

• $|Re(z)| \leq |z|$
• $|Im(z)| \leq |z|$
• $|z|^2 = z \cdot \overline{z} = a^2 + b^2$

2.5 Square Root of a Complex Number

Method 1: Using Direct Formula

For a complex number $z = a + ib$, the square roots are given by:

$$\sqrt{a + ib} = \pm \left[ \sqrt{\frac{|z| + a}{2}} + i \cdot \text{sgn}(b) \sqrt{\frac{|z| - a}{2}} \right]$$

where:

• $|z| = \sqrt{a^2 + b^2}$
• $\text{sgn}(b) = \begin{cases} 1 & \text{if } b > 0 \ -1 & \text{if } b < 0 \end{cases}$

Method 2: Algebraic Method

Let $\sqrt{a + ib} = x + iy$, where $x, y \in \mathbb{R}$

Squaring both sides: $$a + ib = (x + iy)^2 = x^2 - y^2 + 2ixy$$

Equating real and imaginary parts: $$x^2 - y^2 = a \quad \text{...(i)}$$ $$2xy = b \quad \text{...(ii)}$$

Also, $$x^2 + y^2 = \sqrt{a^2 + b^2} = |z| \quad \text{...(iii)}$$

Solving (i) and (iii): $$x^2 = \frac{|z| + a}{2} \quad \text{and} \quad y^2 = \frac{|z| - a}{2}$$

Hence, $$x = \pm \sqrt{\frac{|z| + a}{2}}, \quad y = \pm \sqrt{\frac{|z| - a}{2}}$$

The signs of $x$ and $y$ are determined by equation (ii).

Method 3: Square Root of Pure Imaginary Numbers

For $z = ib$ (where $b > 0$): $$\sqrt{ib} = \pm \sqrt{\frac{b}{2}}(1 + i)$$

For $z = -ib$ (where $b > 0$): $$\sqrt{-ib} = \pm \sqrt{\frac{b}{2}}(1 - i)$$

Examples:

Example 1: Find the square root of $3 + 4i$

Here, $a = 3, b = 4, |z| = \sqrt{9 + 16} = 5$

$$x^2 = \frac{5 + 3}{2} = 4 \Rightarrow x = \pm 2$$ $$y^2 = \frac{5 - 3}{2} = 1 \Rightarrow y = \pm 1$$

Since $2xy = 4 > 0$, $x$ and $y$ have the same sign.

$$\sqrt{3 + 4i} = \pm(2 + i)$$

Example 2: Find the square root of $8 - 6i$

Here, $a = 8, b = -6, |z| = \sqrt{64 + 36} = 10$

$$x^2 = \frac{10 + 8}{2} = 9 \Rightarrow x = \pm 3$$ $$y^2 = \frac{10 - 8}{2} = 1 \Rightarrow y = \pm 1$$

Since $2xy = -6 < 0$, $x$ and $y$ have opposite signs.

$$\sqrt{8 - 6i} = \pm(3 - i)$$

Important Note:

Every non-zero complex number has exactly two square roots which are negatives of each other.

Key Formulas Summary

1. $i^2 = -1$
2. $z = a + ib$ where $a = Re(z)$, $b = Im(z)$
3. $\bar{z} = a - ib$
4. $|z| = \sqrt{a^2 + b^2}$
5. $z \cdot \bar{z} = |z|^2$
6. $\sqrt{a + ib} = \pm \left[ \sqrt{\frac{|z|+a}{2}} + i \cdot \text{sgn}(b) \sqrt{\frac{|z|-a}{2}} \right]$
7. $|z_1 + z_2| \leq |z_1| + |z_2|$
8. $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
9. $z + \bar{z} = 2Re(z)$
10. $z - \bar{z} = 2iIm(z)$

Practice Problems

1. Find the value of $i^{2025}$
2. If $z = 3 - 4i$, find $\bar{z}$ and $|z|$
3. Find the square roots of $5 + 12i$
4. Prove that $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$
5. If $z = \frac{1 + i}{1 - i}$, find $|z|$ and $\bar{z}$
6. Find the square roots of $-7 - 24i$
7. If $|z_1| = 3$ and $|z_2| = 4$, what are the possible values of $|z_1 + z_2|$?
8. Find the square roots of $i$