🌟 Level 4 - Topic 3: Complex Numbers (From Basics to Advanced) 🚀

1) Introduction 🤔

Complex numbers, in the form \( a+bi \) (with \( i^2=-1 \)), are vital in higher-level mathematics, engineering, and physics. They help solve equations that have no real solutions (like \( x^2+1=0 \)), enable us to represent waves and rotations, and appear in signal processing.

Let’s start with the basic concepts (a refresher or introduction for newcomers), then move into advanced techniques like polar/exponential form, De Moivre’s Theorem, roots of complex polynomials, and geometric transformations.

2) Basic Complex Number Operations 🔢

2.1 Rectangular Form

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

  • Real part: \( \Re(z)=a \)
  • Imaginary part: \( \Im(z)=b \)

Addition & Subtraction

\( (a+bi)\pm(c+di)=(a\pm c)+(b\pm d)i \)

Example 1: \((2+3i)+(1-4i)=3-i\).

Multiplication

\( (a+bi)(c+di)=(ac-bd)+(ad+bc)i \)

Example 2: \((1+2i)(3-i)=5+5i\).

Division (Rationalizing Denominator)

\( \frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{(c+di)(c-di)}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2} \)

Example 3: \(\frac{3+2i}{1-i}\)

  1. Multiply numerator and denominator by \((1+i)\).
  2. Numerator: \((3+2i)(1+i)= 3+3i+2i+2i^2=1+5i\) (since \( i^2=-1 \)).
  3. Denominator: \((1-i)(1+i)=1-i^2=2\).
  4. Result: \(\frac{1+5i}{2}=\frac{1}{2}+\frac{5}{2}i\).

3) Polar (Trigonometric) & Exponential Forms ✨

3.1 Polar Form

\( z=r(\cos\theta+i\sin\theta) \)

  • \( r=|z|=\sqrt{a^2+b^2} \)
  • \( \theta=\arg(z)=\tan^{-1}\Bigl(\frac{b}{a}\Bigr) \)

Example 4: If \( z=3+4i \):

  • \( r=\sqrt{3^2+4^2}=5 \)
  • \( \theta\approx\tan^{-1}(\frac{4}{3})\approx53.13^\circ \)

Hence, \( z=5(\cos53.13^\circ+i\sin53.13^\circ) \).

3.2 Exponential (Euler) Form

Using Euler’s formula \( e^{i\theta}=\cos\theta+i\sin\theta \), we can write:

\( z=r\,e^{i\theta} \)

Why? It simplifies multiplication, division, and raising numbers to powers.

4) De Moivre’s Theorem & Powers/Roots 🧩

4.1 Integer Powers (De Moivre)

\( \bigl[r(\cos\theta+i\sin\theta)\bigr]^n=r^n\Bigl(\cos(n\theta)+i\sin(n\theta)\Bigr) \)

Example 5: Express \((2+2i)\) in polar form as \(2\sqrt{2}(\cos45^\circ+i\sin45^\circ)\) and then compute its square using De Moivre’s theorem.

4.2 nth Roots of a Complex Number

The n distinct nth roots of \(z=r(\cos\theta+i\sin\theta)\) are:

\( \sqrt[n]{z}=\sqrt[n]{r}\Bigl[\cos\Bigl(\frac{\theta+2k\pi}{n}\Bigr)+i\sin\Bigl(\frac{\theta+2k\pi}{n}\Bigr)\Bigr],\quad k=0,1,\dots,n-1 \)

Example 6: For \( z=8(\cos60^\circ+i\sin60^\circ) \):

  • \( \sqrt[3]{8}=2 \).
  • There will be 3 cube roots, equally spaced by \(120^\circ\).

5) Geometry & Transformations in the Complex Plane 📐

  • Addition by \( a+bi \) translates the plane.
  • Multiplication by \( re^{i\phi} \) scales by \( r \) and rotates by \(\phi\).
  • Conjugation \( \overline{z}=a-bi \) reflects across the real axis.
  • Magnitude \( |z| \) is the distance from the origin.
  • Argument \(\arg(z)\) is the angle from the positive x-axis.

Example 7: Multiplying any number by \(1+i\) (which has magnitude \(\sqrt{2}\) and angle \(45^\circ\)) rotates it by \(45^\circ\) and scales it by \(\sqrt{2}\).

6) Complex Polynomials & The Fundamental Theorem of Algebra 🔍

6.1 Fundamental Theorem of Algebra

Every degree \( n \) polynomial has exactly \( n \) complex roots (counting multiplicities).

6.2 Factoring Over \(\mathbb{C}\)

Any polynomial can be factored into linear factors \( (x-\alpha) \) with \( \alpha\in\mathbb{C} \). For example, a quadratic with a negative discriminant will have complex roots.

Example 8: \( x^2+1=0 \) implies \( x=\pm i \).

6.3 Roots of Unity

The solutions of \( z^n=1 \) lie on the unit circle and are equally spaced.

7) Additional Advanced Topics (Preview) 🎉

  • Logarithms of Complex Numbers: \( \ln(re^{i\theta})=\ln r+i\theta+2k\pi i \) (multi-valued).
  • Möbius Transformations: \( z\mapsto\frac{az+b}{cz+d} \) (used in advanced geometry).
  • Complex Integration: Contour integrals and residues (key topics in complex analysis).

8) Worked Examples (Detailed) 🍀

Example 9 (Compute a Power)

Problem: Let \( z=\sqrt{2}(\cos45^\circ+i\sin45^\circ) \). Find \( z^5 \).

  1. Magnitude \( r=\sqrt{2} \) and angle \( \theta=45^\circ \).
  2. By De Moivre’s theorem:

    \( z^5=(\sqrt{2})^5\Bigl(\cos(5\times45^\circ)+i\sin(5\times45^\circ)\Bigr) \)

  3. \((\sqrt{2})^5=2^{5/2}=4\sqrt{2}\).
  4. \(5\times45^\circ=225^\circ\), so \(\cos225^\circ=-\frac{\sqrt{2}}{2}\) and \(\sin225^\circ=-\frac{\sqrt{2}}{2}\).
  5. Thus,

    \( z^5=4\sqrt{2}\Bigl(-\frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}\Bigr)=-4-4i \)

Example 10 (Find 4th Roots)

Problem: Find all 4th roots of \(-16\).

  1. Write \(-16\) in polar form: \( -16=16(\cos\pi+i\sin\pi) \).
  2. \(\sqrt[4]{16}=2\).
  3. The 4th roots are:

    \( z_k=2\Bigl[\cos\Bigl(\frac{\pi+2k\pi}{4}\Bigr)+i\sin\Bigl(\frac{\pi+2k\pi}{4}\Bigr)\Bigr],\quad k=0,1,2,3 \)

  4. This yields 4 solutions at angles: \(\frac{\pi}{4},\ \frac{3\pi}{4},\ \frac{5\pi}{4},\ \frac{7\pi}{4}\).

9) Practice Questions 🎯

9.1 Fundamental – Build Skills (10+)

  1. Convert \( z=1-i \) to polar and exponential forms.
  2. Compute \((2+3i)+(4-5i)\) and interpret the result as vector addition.
  3. Multiply \((3+i)(1-2i)\) and simplify.
  4. Divide \(\frac{4+6i}{1-2i}\) and express the answer in \(a+bi\) form.
  5. Use De Moivre’s theorem to compute \( \Bigl(\sqrt{3}(\cos30^\circ+i\sin30^\circ)\Bigr)^4 \).
  6. Find all 3rd roots of \( 8(\cos60^\circ+i\sin60^\circ) \).
  7. If \( z=2e^{i\pi/6} \), compute \( z^2 \) and express in rectangular form.
  8. Solve \( x^2+1=0 \) and write the solutions in rectangular form.
  9. Factor \( x^2+4 \) over \(\mathbb{C}\).
  10. For the circle \( |z-(2+4i)|=5 \), describe its center and radius in the complex plane.

9.2 Challenging – Push Limits (5+)

  1. 🔥 Compute \( (1+i)^6 \) using polar form; compare with direct expansion.
  2. 🔥 Find all 5th roots of \( 32(\cos120^\circ+i\sin120^\circ) \).
  3. 🔥 For a polynomial \( x^3-(2+i)x^2+\dots \), if \( (x-(1+2i)) \) is a factor, find the other factor(s) if possible.
  4. 🔥 Show that if \(\overline{z}=\frac{1}{z}\), then \(|z|^2=1\) (i.e. \(z\) lies on the unit circle).
  5. 🔥 Convert \(\ln\Bigl(4e^{i\pi/3}\Bigr)\) to standard complex form, noting the multi-valued nature of the complex logarithm.

10) Summary

  • We began with basic operations in rectangular form and reviewed polar form.
  • Exponential (Euler) form simplifies multiplication, division, and exponentiation.
  • De Moivre’s theorem makes it easy to compute powers and extract nth roots.
  • Geometric transformations in the complex plane include translation, rotation, scaling, and reflection.
  • Advanced topics—such as complex logarithms, Möbius transformations, and complex integration—offer a gateway to higher-level mathematics and engineering applications.

With these tools, you’re equipped to handle complex numbers from basic arithmetic to advanced applications in math, physics, and engineering. Keep exploring, and enjoy the rich world of complex numbers! 🌟

← Back to Previous Topic