Complex Numbers: Definitions, Operations, Geometry, Polar Form

TAKEAWAYS:

• Complex numbers are numbers of the form $a+bi$, where $i=\sqrt{-1}$, and they can be represented geometrically as points in a plane.
• Operations like addition, subtraction, multiplication, and division follow algebraic rules, but division requires multiplying by the conjugate.
• The polar form $r(\cos \theta +i\sin \theta )$ simplifies multiplication and division using De Moivre’s Theorem.
• Complex numbers solve equations with no real roots and model oscillations, waves, and AC circuits.
• The Argand diagram (complex plane) visualizes complex numbers, while Euler’s formula connects them to trigonometry.
• NEB exams test calculations, proofs, and applications—always show steps clearly and verify answers.

---

What is a Complex Number?

A complex number is a number written in the form: $$z=a+bi$$ where:

• $a$ and $b$ are real numbers,
• $i$ is the imaginary unit, defined as $i=\sqrt{-1}$ (so $i^2=-1$).

Why do we need complex numbers?

Real numbers cannot solve equations like $x^2+1=0$. Complex numbers extend the number system to include solutions like $x=i$ and $x=-i$.

Parts of a Complex Number

For $z=a+bi$:

• $a$ is the real part (Re$(z)$),
• $b$ is the imaginary part (Im$(z)$),
• $i$ is the imaginary unit.

Example 1: Identify the real and imaginary parts. For $z=3+4i$:

• Re$(z)=3$,
• Im$(z)=4$.

For $z=-5i$:

• Re$(z)=0$,
• Im$(z)=-5$.

---

Representing Complex Numbers Geometrically

Complex numbers can be plotted on the complex plane (or Argand diagram), where:

• The x-axis represents the real part,
• The y-axis represents the imaginary part.

Example 2: Plot $z=-1+2i$.

• Move 1 unit left (real part: $-1$),
• Move 2 units up (imaginary part: $2$).

---

Equality of Complex Numbers

Two complex numbers $z_1=a+bi$ and $z_2=c+di$ are equal if and only if: $$a=c\text{and}b=d$$

Example 3: Solve for $x$ and $y$ if $(x+3)+(y-1)i=5-2i$.

• Equate real parts: $x+3=5⟹x=2$,
• Equate imaginary parts: $y-1=-2⟹y=-1$.

---

Operations on Complex Numbers

1. Addition and Subtraction

Add/subtract the real and imaginary parts separately.

Example 4: Compute $(3+4i)+(-2+5i)$. $$(3+(-2))+(4i+5i)=1+9i$$

Example 5: Compute $(5-3i)-(2+7i)$. $$(5-2)+(-3i-7i)=3-10i$$

2. Multiplication

Use the distributive property (FOIL method) and recall $i^2=-1$.

Example 6: Multiply $(2+3i)(1-4i)$. $$=2(1)+2(-4i)+3i(1)+3i(-4i)$$ $$=2-8i+3i-12i^2$$ $$=2-5i-12(-1)(\text{since }{i}^2=-1)$$ $$=2-5i+12=14-5i$$

3. Division

Multiply the numerator and denominator by the conjugate of the denominator to eliminate $i$.

Example 7: Divide $\frac{3+4i}{1-2i}$.

1. Find the conjugate of the denominator: $1+2i$.
2. Multiply numerator and denominator by $1+2i$: $$\frac{(3+4i)(1+2i)}{(1-2i)(1+2i)}$$
3. Expand the numerator: $$=3(1)+3(2i)+4i(1)+4i(2i)=3+6i+4i+8i^2$$ $$=3+10i+8(-1)=3+10i-8=-5+10i$$
4. Expand the denominator (difference of squares): $$(1{)}^2-(2i{)}^2=1-4i^2=1-4(-1)=1+4=5$$
5. Final result: $$\frac{-5+10i}{5}=-1+2i$$

---

The Complex Conjugate

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

Properties:

1. $z+\overline{z}=2a$ (real part),
2. $z-\overline{z}=2bi$ (imaginary part),
3. $z\cdot \overline{z}=a^2+b^2$ (always real and non-negative).

Example 8: Find the conjugate of $5-7i$ and compute $z\cdot \overline{z}$.

• $\overline{z}=5+7i$,
• $z\cdot \overline{z}=(5{)}^2+(7{)}^2=25+49=74$.

---

Modulus and Argument of a Complex Number

Modulus ($|z|$)

The modulus of $z=a+bi$ is: $$|z|=\sqrt{a^2+b^2}$$ It represents the distance from the origin to the point $(a,b)$ in the complex plane.

Example 9: Find $|3+4i|$. $$|3+4i|=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$$

Argument ($\theta$ or $\arg (z)$)

The argument is the angle $\theta$ that the line from the origin to $(a,b)$ makes with the positive x-axis (in radians or degrees).

Example 10: Find $\arg (1-\sqrt{3}i)$.

• Plot $1-\sqrt{3}i$:
  • Real part ($a$) = 1,
  • Imaginary part ($b$) = $-\sqrt{3}$.
• The point lies in the 4th quadrant.
• $\tan \theta =\frac{b}{a}=\frac{-\sqrt{3}}{1}=-\sqrt{3}$,
• $\theta =-\frac{\pi }{3}$ (or $-60^{\circ }$).

---

Polar Form of a Complex Number

A complex number can also be written in polar form as: $$z=r(\cos \theta +i\sin \theta )$$ where:

• $r=|z|=\sqrt{a^2+b^2}$ (modulus),
• $\theta =\arg (z)$ (argument).

Example 11: Convert $z=-1+\sqrt{3}i$ to polar form.

1. Find $r$: $$r=\sqrt{(-1{)}^2+(\sqrt{3}{)}^2}=\sqrt{1+3}=2$$
2. Find $\theta$:
   • The point $(-1,\sqrt{3})$ is in the 2nd quadrant.
   • $\tan \theta =\frac{\sqrt{3}}{-1}=-\sqrt{3}$,
   • Reference angle = $60^{\circ }$,
   • $\theta =180^{\circ }-60^{\circ }=120^{\circ }$ (or $\frac{2\pi }{3}$ radians).
3. Polar form: $$z=2\left(\cos 120^{\circ }+i\sin 120^{\circ }\right)$$

---

Multiplication and Division in Polar Form

Multiplication

Multiply the moduli and add the arguments: $$z_1\cdot z_2=r_1{r}_2\left[\cos ({\theta }_1+{\theta }_2)+i\sin ({\theta }_1+{\theta }_2)\right]$$

Example 12: Multiply $z_1=2(\cos 30^{\circ }+i\sin 30^{\circ })$ and $z_2=3(\cos 45^{\circ }+i\sin 45^{\circ })$. $$z_1\cdot z_2=2\times 3\left[\cos (30^{\circ }+45^{\circ })+i\sin (30^{\circ }+45^{\circ })\right]$$ $$=6(\cos 75^{\circ }+i\sin 75^{\circ })$$

Division

Divide the moduli and subtract the arguments: $$\frac{z_1}{z_2}=\frac{r_1}{r_2}\left[\cos ({\theta }_1-{\theta }_2)+i\sin ({\theta }_1-{\theta }_2)\right]$$

Example 13: Divide $z_1=4(\cos 120^{\circ }+i\sin 120^{\circ })$ by $z_2=2(\cos 30^{\circ }+i\sin 30^{\circ })$. $$\frac{z_1}{z_2}=\frac{4}{2}\left[\cos (120^{\circ }-30^{\circ })+i\sin (120^{\circ }-30^{\circ })\right]$$ $$=2(\cos 90^{\circ }+i\sin 90^{\circ })$$

---

De Moivre’s Theorem

De Moivre’s Theorem states that for any integer $n$: $${\left[r(\cos \theta +i\sin \theta )\right]}^n=r^n(\cos n\theta +i\sin n\theta )$$

Example 14: Find $(1+i{)}^5$ using De Moivre’s Theorem.

1. Convert $1+i$ to polar form:
   • $r=\sqrt{1^2+1^2}=\sqrt{2}$,
   • $\theta ={\tan }^{-1}(1/1)=45^{\circ }$,
   • $1+i=\sqrt{2}(\cos 45^{\circ }+i\sin 45^{\circ })$.
2. Apply De Moivre’s Theorem: $$(1+i{)}^5=(\sqrt{2}{)}^5\left[\cos (5\times 45^{\circ })+i\sin (5\times 45^{\circ })\right]$$ $$=4\sqrt{2}(\cos 225^{\circ }+i\sin 225^{\circ })$$
3. Convert back to rectangular form:
   • $\cos 225^{\circ }=-\frac{\sqrt{2}}{2}$,
   • $\sin 225^{\circ }=-\frac{\sqrt{2}}{2}$,
   • $(1+i{)}^5=4\sqrt{2}\left(-\frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}\right)=-4-4i$.

---

Roots of Complex Numbers

To find the $n$th roots of a complex number $z=r(\cos \theta +i\sin \theta )$, use: $$z_k=r^{1/n}\left[\cos \left(\frac{\theta +2k\pi }{n}\right)+i\sin \left(\frac{\theta +2k\pi }{n}\right)\right],k=0,1,2,\dots ,n-1$$

Example 15: Find the cube roots of $-8$.

1. Express $-8$ in polar form:
   • $-8=8(\cos 180^{\circ }+i\sin 180^{\circ })$,
   • $r=8$, $\theta =180^{\circ }$.
2. Find the cube roots ($n=3$):
   • $r^{1/3}=2$,
   • ${\theta }_k=\frac{180^{\circ }+360^{\circ }k}{3}$, for $k=0,1,2$.
3. Calculate each root:
   • $k=0$: ${\theta }_0=60^{\circ }$, $z_0=2(\cos 60^{\circ }+i\sin 60^{\circ })=2\left(\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)=1+i\sqrt{3}$,
   • $k=1$: ${\theta }_1=180^{\circ }$, $z_1=2(\cos 180^{\circ }+i\sin 180^{\circ })=-2$,
   • $k=2$: ${\theta }_2=300^{\circ }$, $z_2=2(\cos 300^{\circ }+i\sin 300^{\circ })=1-i\sqrt{3}$.

The three cube roots are $1+i\sqrt{3}$, $-2$, and $1-i\sqrt{3}$.

---

Euler’s Formula

Euler’s formula connects complex numbers to trigonometry: $$e^{i\theta }=\cos \theta +i\sin \theta$$ Thus, a complex number can also be written as: $$z=r{e}^{i\theta }$$

Example 16: Express $z=\sqrt{3}+i$ in exponential form.

1. Find $r$ and $\theta$:
   • $r=\sqrt{(\sqrt{3}{)}^2+1^2}=2$,
   • $\theta ={\tan }^{-1}(1/\sqrt{3})=30^{\circ }$,
2. Exponential form: $$z=2e^{i\pi /6}$$

---

Applications of Complex Numbers

1. Solving Quadratic Equations: Complex numbers solve equations like $x^2+4=0$ (roots: $x=\pm 2i$).
2. Electrical Engineering: Model AC circuits using impedance (resistance + reactance).
3. Signal Processing: Represent waves and oscillations (e.g., Fourier transforms).
4. Fractals: Used in computer graphics (e.g., Mandelbrot set).

---

Summary Table: Operations on Complex Numbers

---

Exam Tip

1. Show all steps: NEB exams deduct marks for missing work. For example, when dividing complex numbers, always write:
   • "Multiply numerator and denominator by the conjugate of the denominator."
2. Verify answers: After solving, check if your answer makes sense (e.g., modulus is non-negative, argument is in the correct quadrant).
3. Polar form is key: For multiplication/division, polar form is easier. Convert to polar if the numbers are given in rectangular form.
4. Graphical understanding: Always plot complex numbers to visualize operations (e.g., multiplication rotates and scales the point).
5. Common mistakes:
   • Forgetting $i^2=-1$ in multiplication.
   • Incorrectly calculating the argument (e.g., missing quadrant adjustments).
   • Skipping the conjugate step in division.
6. Practice: Solve at least 5 problems for each operation (addition, multiplication, division, modulus, argument, polar form).

---

NEB Board-Style Questions

Short Answer (5 marks each)

1. Addition/Subtraction: Compute $(4-3i)+(2+5i)-(1-i)$.

2. Multiplication: Multiply $(1+2i)(3-4i)$ and verify using polar form.

3. Division: Divide $\frac{5+6i}{2-3i}$ and express the answer in standard form.

4. Modulus and Argument: For $z=-3+3i$, find $|z|$ and $\arg (z)$ in radians.

5. Polar Form: Convert $z=1-\sqrt{3}i$ to polar form and find its square using De Moivre’s Theorem.

Long Answer (10 marks each)

6. Roots of Unity: Find all the cube roots of unity and represent them geometrically on the complex plane.

7. Equation Solving: Solve $x^2+x+1=0$ using complex numbers and express the roots in polar form.

8. Application: An AC circuit has impedance $Z=3+4i$ ohms and current $I=2-i$ amperes. Find the voltage $V=I\cdot Z$ in rectangular form.

---

Hint: For long answers, always:

• Start with the given information,
• Show every algebraic step,
• Interpret results (e.g., "The roots lie on a circle in the complex plane").