Basic MathematicsUnit 18 min read

Complex Numbers – Rectangular & Polar Forms, Roots, De Moivre’s Theorem

Unit 1 of Basic Mathematics introduces complex numbers, their algebraic and polar representations, computation of powers and roots using De Moivre’s theorem, and real‑world applications in engineering and digital services.

Padhai · Based on the official syllabus and past papers
Updated 5 Oct 2026

Key points

Complex numbers can be written uniquely in rectangular $(a+ib)$ and polar $(r\cis\theta)$ forms.
Multiplication, division, powers and roots become simple angle‑and‑modulus operations in polar form.
De Moivre’s theorem provides a fast way to raise a complex number to any integer power and to extract $n$th roots.
Argand diagrams visualise complex numbers as points/vectors in a plane, aiding geometric intuition.
Real‑world systems such as signal processing, antenna design and cryptographic algorithms rely on complex arithmetic.

1. What is a Complex Number?

A complex number is an ordered pair of real numbers written as

$z = a + ib, a, b \in R, i^{2} = - 1$

Rectangular (Cartesian) form: $a$ is the real part $Re (z)$ , $b$ is the imaginary part $Im (z)$ .
Geometric view: the point $(a, b)$ in the complex (Argand) plane.

1.1 Modulus and Argument

Modulus (distance from origin):

$∣ z ∣ = r = a^{2} + b^{2}$

Argument (counter‑clockwise angle from positive real axis):

$θ = arg (z) = tan^{- 1} (\frac{b}{a})$

The pair $(r, θ)$ uniquely determines $z$ .

1.2 Polar (Trigonometric) Form

$z = r (cos θ + i sin θ) = r \cis θ$

Euler’s formula gives the exponential version

$z = r e^{i θ}$

Conversion Table

Form	Expression	How to obtain
Rectangular	$a + ib$	Given directly
Polar	$r \cis θ$	$r = a^{2} + b^{2}, θ = tan^{- 1} (b / a)$
Exponential	$r e^{i θ}$	Same $r, θ$ as polar

2. Algebra with Complex Numbers

2.1 Addition & Subtraction (Rectangular)

$(a + ib) \pm (c + i d) = (a \pm c) + i (b \pm d)$

2.2 Multiplication (Polar)

If $z_{1} = r_{1} \cis θ_{1}$ and $z_{2} = r_{2} \cis θ_{2}$ ,

$z_{1} z_{2} = r_{1} r_{2} \cis (θ_{1} + θ_{2})$

2.3 Division (Polar)

$\frac{z _{1}}{z _{2}} = \frac{r _{1}}{r _{2}} \cis (θ_{1} - θ_{2}), z_{2} \neq = 0$

2.4 Conjugate

$\overline{z} = a - ib, ∣ z ∣^{2} = z \overline{z}$

3. Powers and Roots – De Moivre’s Theorem

3.1 Statement

For any integer $n$ ,

$(r \cis θ)^{n} = r^{n} \cis (n θ)$

3.2 Computing Powers

Example 3.1 – Compute $(1 + i)^{5}$ .

Convert to polar:
$r = 1^{2} + 1^{2} = 2, θ = tan^{- 1} (1) = \frac{π}{4}$
Apply De Moivre:
$(1 + i)^{5} = (2)^{5} \cis (5 \cdot \frac{π}{4}) = 42 \cis (\frac{5 π}{4})$
Return to rectangular:
$42 (cos \frac{5 π}{4} + i sin \frac{5 π}{4}) = 42 (- \frac{2}{2} - i \frac{2}{2}) = - 4 - 4 i$

3.3 Extracting $n$ th Roots

If $z = r \cis θ$ , the $n$ distinct roots are

$z_{k} = r^{1/ n} \cis (\frac{θ + 2 k π}{n}), k = 0, 1, \dots, n - 1$

Worked Example 3.2 – Find the square roots of $z = 1 + i 3$ .

Polar form of $z$
$a = 1, b = 3 \Rightarrow r = 1 + 3 = 2, θ = tan^{- 1} (\frac{3}{1}) = \frac{π}{3}$
Root modulus: $r^{1/2} = 2$ .
Root arguments:

$θ_{0} = \frac{π /3}{2} = \frac{π}{6}, θ_{1} = \frac{π /3 + 2 π}{2} = \frac{7 π}{6}$

Roots in rectangular form

$z_{0} z_{1} = 2 \cis \frac{π}{6} = 2 (cos \frac{π}{6} + i sin \frac{π}{6}) = 2 (\frac{3}{2} + i \frac{1}{2}) = \frac{6}{2} + i \frac{2}{2} = 2 \cis \frac{7 π}{6} = 2 (cos \frac{7 π}{6} + i sin \frac{7 π}{6}) = 2 (- \frac{3}{2} - i \frac{1}{2}) = - \frac{6}{2} - i \frac{2}{2}$

4. Common Mistakes & How to Avoid Them

Mistake	Why it Happens	Correct Approach
Forgetting to add $2 k π$ when finding roots	Only one angle considered	Always write $θ + 2 k π$ before dividing by $n$
Mixing degrees and radians	Calculator set to wrong mode	Keep a consistent unit; exam prefers radians
Using rectangular form for powers >2	Algebra becomes messy	Convert to polar first, apply De Moivre, then back‑convert
Mis‑identifying the principal argument	$tan^{- 1}$ returns values in $(- π /2, π /2)$ only	Adjust quadrant based on signs of $a, b$

5. Applications of Complex Numbers

Application	How Complex Numbers Appear	Example
Digital Signal Processing (DSP) – FFT	FFT algorithm manipulates complex exponentials to transform time‑domain signals to frequency domain.	YouTube video compression uses FFT to analyse audio frequencies.
Wireless Communications – Phased‑array antennas	Complex phasors represent amplitude and phase of each antenna element.	Ncell’s 4G base stations steer beams using complex phasor sums.
Cryptography (RSA, ECC)	While RSA uses modular arithmetic, elliptic‑curve cryptography (ECC) works over finite fields that can be visualised with complex‑like geometry.	eSewa’s secure transaction layer employs ECC for key exchange.
Computer Graphics – Rotations	Rotating a point $(x, y)$ by angle $θ$ is equivalent to multiplying the complex number $x + i y$ by $e^{i θ}$ .	Daraz’s product‑image carousel rotates thumbnails smoothly.

Worked Real‑World Example

A bank offers a loan with annual interest rate $i = 12%$ compounded continuously. The growth factor after $t$ years is $e^{i t}$ . If the bank models the interest as a complex exponential to handle periodic promotional adjustments, the effective factor after 1 year with a seasonal adjustment of phase $ϕ = π /6$ becomes

$e^{i (0.12 + ϕ)} = e^{i 0.12} e^{iπ /6} = (cos 0.12 + i sin 0.12) (cos \frac{π}{6} + i sin \frac{π}{6})$

Evaluating the product uses the same multiplication rule as for complex numbers, illustrating that the same algebraic machinery underlies both financial modelling and signal processing.

6. In the Real World

YouTube uses the Fast Fourier Transform (FFT) to compress audio streams. FFT treats each audio sample as a complex number; the magnitude gives frequency strength, the argument gives phase.
Ncell designs its 4G antenna arrays with complex phasors. Each antenna element’s signal is represented as $A e^{i θ}$ ; the combined field is the vector sum, directly applying addition of complex numbers.
Daraz implements image rotation on the client side using the formula $(x + i y) \times e^{i θ}$ . When a user swipes to view a product from a different angle, the browser multiplies the pixel coordinate complex number by a unit‑modulus complex number, achieving smooth rotation without trigonometric function calls for each pixel.

7. Summary of Key Formulas

$z r z_{1} z_{2} \frac{z _{1}}{z _{2}} z^{n} n z_{k} = a + ib = r \cis θ = a^{2} + b^{2}, θ = tan^{- 1} \frac{b}{a} = r_{1} r_{2} \cis (θ_{1} + θ_{2}) = \frac{r _{1}}{r _{2}} \cis (θ_{1} - θ_{2}) = r^{n} \cis (n θ) (De Moivre) = r^{1/ n} \cis (\frac{θ + 2 k π}{n}), k = 0, \dots, n - 1$

Exam tip

Always convert to polar before raising to a power or extracting roots – the exam frequently gives a rectangular number and asks for $(z)^{5}$ or $n z$ .
Write the argument in radians; the standard answer expects $θ = \frac{π}{3}, \frac{5 π}{4}$ etc.
For roots, list all $n$ values of $k$ – missing one root loses marks.
Show the Argand diagram if the question asks for a geometric interpretation; a quick sketch of the vectors earns partial credit.
Check the principal argument: after computing $tan^{- 1} (b / a)$ , adjust to the correct quadrant before applying De Moivre.

Based on the TU BITM syllabus for Basic Mathematics (MTH204), unit 1.

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Complex Numbers – Rectangular & Polar Forms, Roots, De Moivre’s Theorem