Complex Numbers — Revision Notes

1. Fundamentals of Complex Numbers

Definition: A number of the form $z = a + ib$ where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$ (iota).
Parts:
Real Part: $\text{Re}(z) = a$
Imaginary Part: $\text{Im}(z) = b$
Powers of Iota ($i$):
$i = \sqrt{-1}$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

2. Basic Operations

For $z_1 = a + ib$ and $z_2 = c + id$:

Addition: $z_1 + z_2 = (a+c) + i(b+d)$
Subtraction: $z_1 - z_2 = (a-c) + i(b-d)$
Multiplication: $z_1 \cdot z_2 = (ac - bd) + i(ad + bc)$
Division: $\frac{z_1}{z_2} = \frac{(a+ib)(c-id)}{c^2+d^2} = \left(\frac{ac+bd}{c^2+d^2}\right) + i\left(\frac{bc-ad}{c^2+d^2}\right)$
Equality: $a + ib = x + iy \implies a = x$ and $b = y$

3. Conjugate, Modulus & Inverse

Conjugate ($\overline{z}$):
If $z = a + ib$, then $\overline{z} = a - ib$.
Key Property: $z \cdot \overline{z} = a^2 + b^2$
Modulus ($|z|$):
$|z| = \sqrt{a^2 + b^2}$ (distance of $z$ from origin in the Argand plane)
Multiplicative Inverse ($z^{-1}$):
$z^{-1} = \frac{1}{z} = \frac{\overline{z}}{|z|^2} = \frac{a}{a^2+b^2} + i\left(\frac{-b}{a^2+b^2}\right)$

4. Argument and Quadrant Rules

Reference angle: $\alpha = \tan^{-1}\left|\frac{y}{x}\right|$

Quadrant	Location of $(x, y)$	Principal Argument ($\theta$)
1st Quadrant	$x > 0, y > 0$	$\theta = \alpha$
2nd Quadrant	$x < 0, y > 0$	$\theta = 180^\circ - \alpha = \pi - \alpha$
3rd Quadrant	$x < 0, y < 0$	$\theta = -180^\circ + \alpha = -\pi + \alpha$
4th Quadrant	$x > 0, y < 0$	$\theta = -\alpha$

5. Polar Form & De Moivre's Theorem

Polar Form: $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ and $\theta = \text{arg}(z)$.
De Moivre's Theorem: For any integer $n$: $$z^n = [r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$$

6. Square Root of a Complex Number

To find $\sqrt{x + iy} = a + ib$: 1. Square both sides: $x + iy = (a^2 - b^2) + i(2ab)$ 2. Equate parts: $a^2 - b^2 = x$ and $2ab = y$ 3. Use identity: $a^2 + b^2 = \sqrt{(a^2 - b^2)^2 + 4a^2b^2} = \sqrt{x^2 + y^2}$ 4. Solve for $a$ and $b$, maintaining correct sign alignment from $2ab$

Example:

Q. Find Square Root of: $16-i30$

$sol^n$: Let, $\sqrt{16-i30}=a+ib$

Sqring both sides: $\implies$ $16-i30=(a+ib)^2$ $\implies$ $16-i30=(a^2-b^2)+i(2b)$

By comparing LHS with RHS, We Get:

$\implies\boxed{(a^2-b^2)=16}\rightarrow(1)$

$\implies\boxed{2b=30}\rightarrow(2)$

By sqring and adding $(1)$ and $(2)$ ,

$(a^2-b^2)+(2b)^2=(16)^2+(30)^2$

$\implies(a^2+b^2)=256+900$

$\implies a^2+b^2=\sqrt{1156}$

$\implies \boxed{a^2+b^2=34}\rightarrow(3)$

7. Solved Example Summary

Problem Type	Given Expression	Result / Solution
Simplification	$\frac{1+i}{1-i}$	$0 + i(1) = i$
Multiplicative Inverse	$7 - 2i$	$\frac{7}{53} + i\left(\frac{2}{53}\right)$
Square Root	$\sqrt{8 - 15i}$	$\pm\left(\frac{5}{\sqrt{2}} - i\frac{3}{\sqrt{2}}\right)$
Modulus & Argument	$1 + i\sqrt{3}$	$r = 2$, $\theta = \frac{\pi}{3}\ (60^\circ)$
De Moivre's Power	$(1+i)^{16}$	$(\sqrt{2})^{16}(\cos 4\pi + i\sin 4\pi) = 256$

Quadrant	Location of \((x, y)\)	Principal Argument (\(\theta\))
1st Quadrant	\(x > 0, y > 0\)	\(\theta = \alpha\)
2nd Quadrant	\(x < 0, y > 0\)	\(\theta = 180^\circ - \alpha = \pi - \alpha\)
3rd Quadrant	\(x < 0, y < 0\)	\(\theta = -180^\circ + \alpha = -\pi + \alpha\)
4th Quadrant	\(x > 0, y < 0\)	\(\theta = -\alpha\)

Problem Type	Given Expression	Result / Solution
Simplification	\(\frac{1+i}{1-i}\)	\(0 + i(1) = i\)
Multiplicative Inverse	\(7 - 2i\)	\(\frac{7}{53} + i\left(\frac{2}{53}\right)\)
Square Root	\(\sqrt{8 - 15i}\)	\(\pm\left(\frac{5}{\sqrt{2}} - i\frac{3}{\sqrt{2}}\right)\)
Modulus & Argument	\(1 + i\sqrt{3}\)	\(r = 2\), \(\theta = \frac{\pi}{3}\ (60^\circ)\)
De Moivre's Power	\((1+i)^{16}\)	\((\sqrt{2})^{16}(\cos 4\pi + i\sin 4\pi) = 256\)