1
Theory
Definition: A complex number $z = x + iy$, where $x \in \mathbb{R}$ (real part) and $y \in \mathbb{R}$ (imaginary part), and $i = \sqrt{-1}$ with $i^2 = -1$.
Polar Form: $z = r(\cos\theta + i\sin\theta)$, where $r = |z| = \sqrt{x^2 + y^2}$ (modulus) and $\theta = \arg(z) = \tan^{-1}(y/x)$ (argument).
Euler's Form: $z = re^{i\theta}$, where $e^{i\theta} = \cos\theta + i\sin\theta$. Connects exponential and trigonometric representations.
Conjugate: If $z = x + iy$, then $\bar{z} = x - iy$. Key property: $z \cdot \bar{z} = |z|^2$. Also $z + \bar{z} = 2\text{Re}(z)$ and $z - \bar{z} = 2i\text{Im}(z)$.
Modulus Properties: $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$, $|z_1/z_2| = |z_1|/|z_2|$, Triangle Inequality: $||z_1|-|z_2|| \leq |z_1 \pm z_2| \leq |z_1| + |z_2|$.
De Moivre's Theorem: $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$ for any rational $n$. Essential for finding powers and roots.
nth Roots of Unity: Solutions of $z^n = 1$ are $z_k = e^{2\pi ik/n} = \cos(2\pi k/n) + i\sin(2\pi k/n)$ for $k = 0, 1, \ldots, n-1$. Sum of all roots $= 0$ (for $n > 1$).
Locus: $|z - z_0| = r$ → Circle (center $z_0$, radius $r$); $|z - z_1| = |z - z_2|$ → Perpendicular bisector; $\arg(z - z_0) = \alpha$ → Ray from $z_0$ at angle $\alpha$.
Rotation: Multiplying $z$ by $e^{i\alpha}$ rotates $z$ counter-clockwise by angle $\alpha$ about origin without changing magnitude.
Important Values: $i^{4n}=1$, $i^{4n+1}=i$, $i^{4n+2}=-1$, $i^{4n+3}=-i$. Also $(1+i)^2 = 2i$, $(1-i)^2 = -2i$, $\frac{1+i}{1-i}=i$.
2
Formula Sheet
$$\begin{aligned}
&\textbf{Basic Operations:} && z_1 \pm z_2 = (x_1 \pm x_2) + i(y_1 \pm y_2) \\[6pt]
& && z_1 \cdot z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1) \\[6pt]
& && \frac{z_1}{z_2} = \frac{z_1 \cdot \bar{z}_2}{|z_2|^2} \quad (z_2 \neq 0) \\[12pt]
&\textbf{Modulus \& Argument:} && |z| = \sqrt{x^2 + y^2}, \quad \arg(z) = \tan^{-1}\!\left(\frac{y}{x}\right) \\[6pt]
& && |z|^2 = z \cdot \bar{z}, \quad \arg(\bar{z}) = -\arg(z) \\[12pt]
&\textbf{De Moivre's Theorem:} && (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) \\[12pt]
&\textbf{Euler's Formula:} && e^{i\theta} = \cos\theta + i\sin\theta, \quad e^{-i\theta} = \cos\theta - i\sin\theta \\[6pt]
& && \cos\theta = \frac{e^{i\theta}+e^{-i\theta}}{2}, \quad \sin\theta = \frac{e^{i\theta}-e^{-i\theta}}{2i} \\[12pt]
&\textbf{nth Roots of } z = r(\cos\theta + i\sin\theta): && z_k = r^{1/n}\!\left[\cos\!\left(\frac{\theta+2k\pi}{n}\right) + i\sin\!\left(\frac{\theta+2k\pi}{n}\right)\right] \\[12pt]
&\textbf{Triangle Inequality:} && ||z_1|-|z_2|| \leq |z_1 \pm z_2| \leq |z_1| + |z_2| \\[12pt]
&\textbf{Cube Roots of Unity:} && 1 + \omega + \omega^2 = 0, \quad \omega^3 = 1, \quad \omega^2 = \bar{\omega}
\end{aligned}$$
3
Solved Examples
Problem 1
Modulus & Argument
Q. Find the modulus and principal argument of $z = \dfrac{(1+i)^3(1+\sqrt{3}i)}{(1-i\sqrt{3})(-1+i)}$
📝 Solution
Converting to polar form:$1+i = \sqrt{2}\,e^{i\pi/4}$, so $(1+i)^3 = 2\sqrt{2}\,e^{i3\pi/4}$
$1+\sqrt{3}i = 2\,e^{i\pi/3}$
$1-i\sqrt{3} = 2\,e^{-i\pi/3}$
$-1+i = \sqrt{2}\,e^{i3\pi/4}$
$$z = \frac{2\sqrt{2} \cdot 2}{2 \cdot \sqrt{2}} \cdot \frac{e^{i(3\pi/4 + \pi/3)}}{e^{i(-\pi/3 + 3\pi/4)}} = 2 \cdot e^{i(2\pi/3)}$$
$|z| = 2$, $\arg(z) = \dfrac{2\pi}{3}$
Problem 2
Locus Problem
Q. If $|z-1| + |z+1| = 4$, find the maximum value of $|z|$.
📝 Solution
The equation represents an ellipse with foci at $z = 1$ and $z = -1$.Sum of distances $= 2a = 4 \Rightarrow a = 2$
Distance between foci $= 2c = 2 \Rightarrow c = 1$
Using $b^2 = a^2 - c^2$: $b^2 = 4 - 1 = 3$
Maximum $|z|$ occurs at vertex on major axis: $\max|z| = a = 2$
Verification: At $z = 2$: $|2-1| + |2+1| = 1 + 3 = 4$ ✓
Maximum value of $|z| = 2$
Problem 3
Roots of Unity
Q. Find the value of $\omega^{2023} + \omega^{2024} + \omega^{2025}$ where $\omega$ is a non-real cube root of unity.
📝 Solution
Since $\omega$ is a non-real cube root of unity: $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$Reducing exponents modulo 3:
• $2023 = 3 \times 674 + 1 \Rightarrow \omega^{2023} = \omega$
• $2024 = 3 \times 674 + 2 \Rightarrow \omega^{2024} = \omega^2$
• $2025 = 3 \times 675 + 0 \Rightarrow \omega^{2025} = 1$
$$\omega^{2023} + \omega^{2024} + \omega^{2025} = \omega + \omega^2 + 1 = 0$$
Answer: $0$
4
Multiple Choice Questions
Single Correct (Questions 1–5)
Q1. If $z = \dfrac{1+7i}{(2-i)^2}$, then $\arg(z)$ equals: Single
Q2. The least positive integer $n$ for which $\left(\dfrac{1+i}{1-i}\right)^n$ is real, is: Single
Q3. If $|z - 3 + 2i| \leq 4$, then the difference between the maximum and minimum values of $|z + 1 + 2i|$ is: Single
Q4. If $|z_1| < 1 < |z_2|$, then $z - z_1 = \lambda(z - z_2)$ ($\lambda \in \mathbb{R}, \lambda \neq 1$) represents: Single
Q5. $\displaystyle\sum_{r=1}^{11} \left[\sin\left(\dfrac{2r\pi}{11}\right) + i\cos\left(\dfrac{2r\pi}{11}\right)\right]$ equals: Single
Multiple Correct (Questions 6–8)
Q6. Which of the following statements are TRUE? Multi
Q7. Let $\omega$ be a non-real cube root of unity. Which of the following equal zero? Multi
Q8. If $z = x + iy$ satisfies $|z + 4| = |z - 4|$, then: Multi
5
Numerical Problems
Numerical N1
If $z = \cos\theta + i\sin\theta$, find the value of $\displaystyle\sum_{m=1}^{15} \text{Im}(z^m)$ when $\theta = \dfrac{2\pi}{15}$. (Enter integer answer)
💡 Hint: Use geometric series and properties of roots of unity.
Numerical N2
Let $z_k$ ($k = 0, 1, 2, \ldots, 9$) be all the 10th roots of unity. Find $\displaystyle\prod_{k=0}^{9} (2 - z_k)$. (Enter integer answer)
💡 Hint: Use the fact that $\prod_{k=0}^{n-1}(x - z_k) = x^n - 1$.