Complex Numbers – definitions, arithmetic, polar form, roots, applications

1. What is a Complex Number?

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

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

• Real part: $\mathrm{Re}(z)=a$
• Imaginary part: $\mathrm{Im}(z)=b$

Geometrically, $z$ corresponds to the point $(a,b)$ or the vector from the origin to that point in the Argand plane.

Figure 1 – Argand diagram of $z=3+4i$. The blue arrow represents the complex number as a vector.

---

2. Basic Operations

Worked Example 1 – Multiplication

Compute $(3+2i)(1-4i)$.

$$\begin{array}{rl}(3+2i)(1-4i)& =3\cdot 1+3(-4i)+2i\cdot 1+2i(-4i)\\ & =3-12i+2i-8i^2\\ & =3-10i-8(-1)\\ & =3-10i+8=11-10i.\end{array}$$

The product $11-10i$ lies in quadrant IV of the Argand plane, confirming the rotation effect.

---

3. Modulus and Argument

• Modulus (absolute value): $|z|=\sqrt{a^2+b^2}$ – distance from the origin.
• Argument: $\arg (z)=\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$ – angle measured from the positive real axis (principal value $-\pi <\theta \le \pi$).

Figure 2 – Polar representation of $z=3+4i$: radius $r=5$, angle $\theta \approx 53^{\circ }$.

Worked Example 2 – Modulus & Argument

For $z=-1+i$:

$$|z|=\sqrt{(-1{)}^2+1^2}=\sqrt{2},\theta ={\tan }^{-1}\left(\frac{1}{-1}\right)={\tan }^{-1}(-1)=-\frac{\pi }{4}(\text{or }135^{\circ }\text{ in QII}).$$

Thus $z=\sqrt{2}(\cos 135^{\circ }+i\sin 135^{\circ })$.

---

4. Polar (Trigonometric) Form

Any non‑zero complex number can be written as

$$z=r(\cos \theta +i\sin \theta )=r\mathrm{cis}\theta ,$$

where $r=|z|$ and $\theta =\arg (z)$.

Conversion Steps (Mermaid flowchart)

Advantages of Polar Form

• Multiplication: $z_1{z}_2=r_1{r}_2\mathrm{cis}({\theta }_1+{\theta }_2)$
• Division: $\frac{z_1}{z_2}=\frac{r_1}{r_2}\mathrm{cis}({\theta }_1-{\theta }_2)$
• Powers: $z^n=r^n\mathrm{cis}(n\theta )$ (De Moivre)

---

5. De Moivre’s Theorem

For integer $n$,

$$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta ).$$

Application: Computing $(1+i{)}^{10}$.

1. Convert $1+i$ to polar: $|1+i|=\sqrt{2},\theta =45^{\circ }$.
2. Apply De Moivre:

$$(1+i{)}^{10}=(\sqrt{2}{)}^{10}\mathrm{cis}(10\cdot 45^{\circ })=2^5\mathrm{cis}(450^{\circ })=32\mathrm{cis}(90^{\circ })=32i.$$

---

6. $n$th Roots of a Complex Number

The equation $z^n=w$ (with $w\ne 0$) has exactly $n$ distinct solutions:

$$z_k=\sqrt[n]{R}\mathrm{cis}\left(\frac{\Phi +2k\pi }{n}\right),k=0,1,\dots ,n-1,$$

where $w=R\mathrm{cis}\Phi$.

Worked Example 3 – Cube Roots of $8$

Write $8=8\mathrm{cis}0$.

$$z_k=\sqrt[3]{8}\mathrm{cis}\left(\frac{0+2k\pi }{3}\right)=2\mathrm{cis}\left(\frac{2k\pi }{3}\right),k=0,1,2.$$

Thus the roots are

$$\begin{array}{rl}{z}_0& =2\mathrm{cis}0=2,\\ z_1& =2\mathrm{cis}\frac{2\pi }{3}=2(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)=-1+\sqrt{3}i,\\ z_2& =2\mathrm{cis}\frac{4\pi }{3}=2(-\frac{1}{2}-\frac{\sqrt{3}}{2}i)=-1-\sqrt{3}i.\end{array}$$

Figure 3 – The three cube roots of 8 are equally spaced on a circle of radius 2.

---

7. Complex Conjugate and Its Properties

The conjugate of $z=a+bi$ is $\overline{z}=a-bi$.

Key properties:

• $z\overline{z}=|z{|}^2$.
• $\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}$.
• $\overline{z_1{z}_2}=\overline{z_1}\overline{z_2}$.

Worked Example 4 – Absolute Value via Conjugate

Find $|\frac{3-4i}{3+4i}|$.

$$\begin{array}{rl}\left|\frac{3-4i}{3+4i}\right|& =\frac{|3-4i|}{|3+4i|}\\ & =\frac{\sqrt{3^2+(-4{)}^2}}{\sqrt{3^2+4^2}}=\frac{5}{5}=1.\end{array}$$

Thus the modulus is 1, meaning the fraction lies on the unit circle.

---

8. Applications of Complex Numbers

Real‑World Worked Example – Phasor Addition in Ncell

Suppose two sinusoidal signals arrive at a base‑station:

• Signal A: amplitude 5 V, phase $30^{\circ }$.
• Signal B: amplitude 3 V, phase $-45^{\circ }$.

Represent as phasors:

$$Z_A=5\mathrm{cis}30^{\circ },Z_B=3\mathrm{cis}(-45^{\circ }).$$

Add in rectangular form:

$$\begin{array}{rl}{Z}_A& =5(\cos 30^{\circ }+i\sin 30^{\circ })=5\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)=\frac{5\sqrt{3}}{2}+\frac{5}{2}i,\\ Z_B& =3(\cos (-45^{\circ })+i\sin (-45^{\circ }))=3\left(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i\right)=\frac{3\sqrt{2}}{2}-\frac{3\sqrt{2}}{2}i.\end{array}$$

Summing:

$$Z_{\text{total}}=\left(\frac{5\sqrt{3}}{2}+\frac{3\sqrt{2}}{2}\right)+i\left(\frac{5}{2}-\frac{3\sqrt{2}}{2}\right).$$

Modulus and argument give the resultant amplitude and phase, which the hardware uses to adjust antenna tuning.

---

9. Common Mistakes & How to Avoid Them

---

10. Summary Table

| Concept | Rectangular Form | Polar Form | Key Formula |
|---------|------------------|------------|-------------|
| Number | \(a+bi\) | \(r\operatorname{cis}\theta\) | \(r=\sqrt{a^{2}+b^{2}},\ \theta=\tan^{-1}(b/a)\) |
| Conjugate | \(\overline{z}=a-bi\) | \(\overline{z}=r\operatorname{cis}(-\theta)\) | \(\;z\overline{z}=r^{2}\) |
| Multiplication | \((a+bi)(c+di)\) | \(r_1r_2\operatorname{cis}(\theta_1+\theta_2)\) | — |
| Division | \(\frac{z_1}{z_2}= \frac{z_1\overline{z_2}}{|z_2|^{2}}\) | \(\frac{r_1}{r_2}\operatorname{cis}(\theta_1-\theta_2)\) | — |
| Power | — | \(r^{n}\operatorname{cis}(n\theta)\) | De Moivre |
| Roots | — | \(\sqrt[n]{R}\operatorname{cis}\bigl(\frac{\Phi+2k\pi}{n}\bigr)\) | — |

---

11. In the real world

• Ncell – Phasor analysis: The base‑station’s signal‑processing unit treats each carrier as a complex phasor; multiplication by $e^{i\theta }$ rotates the signal to the required phase, exactly the polar multiplication rule.
• eSewa – RSA encryption: While RSA itself uses modular arithmetic, the underlying proof of security employs Euler’s theorem, which can be visualised using complex exponentials on the unit circle.
• Daraz – Image rotation: Rotating a product image by $\theta$ degrees is performed by converting each pixel coordinate $(x,y)$ to a complex number $z=x+iy$ and computing $z^{\prime }=z\mathrm{cis}\theta$. The resulting coordinates are plotted back onto the screen.

---

12. Exam tip

1. Always convert to polar form before multiplying, dividing, or raising to a power – it reduces the algebra to simple addition/subtraction of angles.
2. For roots, write the given number in polar form, then apply the root formula; remember to list all $n$ solutions with the $2k\pi$ term.
3. When a question asks for the modulus of a quotient, use $|z_1/z_2|=|z_1|/|z_2|$ – no need to compute the full fraction.
4. Mark the Argand diagram if the problem involves geometric interpretation; a quick sketch of the vectors often reveals the answer (e.g., checking orthogonality via dot product $a_1{a}_2+b_1{b}_2=0$).
5. Watch the sign of the argument: use the correct quadrant (atan2 function) to avoid a 180° error, especially for numbers with negative real parts.

---