1) Introduction to Complex Numbers 🤔
At this expert level, we revisit complex numbers (numbers of the form \( a+bi \) where \( i^2=-1 \)) and explore more sophisticated techniques and applications:
- Polar & exponential forms
- De Moivre’s theorem
- Roots of complex polynomials
- Advanced operations & problem-solving
Complex numbers appear in engineering (signal processing, electrical circuits), physics (wave mechanics), and higher mathematics (advanced equations, transformations).
Please note this is an optional section, as you need to be familiar with Trigonometric froms. You can refer back to this section once you learn that
2) Algebra of Complex Numbers 🔢
2.1 Basic Form
Any complex number \( z \) can be written as:
\( z = a+bi \)
- \( a \) = real part, \( \Re(z) \)
- \( b \) = imaginary part, \( \Im(z) \)
- \( i = \sqrt{-1} \)
Addition/Subtraction:
\( (a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i \)
Multiplication:
\( (a+bi)(c+di) = (ac-bd) + (ad+bc)i \)
Division (rationalizing):
\( \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)} = \frac{(ac+bd) + (bc-ad)i}{c^2+d^2} \)
Example 1 (Compute):
\( (2+3i) + (1-5i) = 3-2i \)
Example 2 (Multiply):
\( (2+i)(3-4i) = 10-5i \)
(Recall that \( i^2=-1 \)).
3) Polar (Trigonometric) Form 📐
A complex number \( z=a+bi \) can also be represented in polar form:
\( z = r(\cos\theta + i\sin\theta) \)
- \( r = |z| = \sqrt{a^2+b^2} \) (the magnitude)
- \( \theta = \arg(z) = \tan^{-1}\Bigl(\frac{b}{a}\Bigr) \) (the angle)
Example 3 (Convert to Polar):
\( z=3+4i \) → \( r=\sqrt{3^2+4^2}=5,\quad \theta\approx\tan^{-1}(4/3)\approx0.9273\; \text{radians} \)
Thus, \( z=5\Bigl(\cos(0.9273)+i\sin(0.9273)\Bigr) \).
4) Exponential Form & Euler’s Formula 🌍
Euler’s formula states:
\( e^{i\theta} = \cos\theta + i\sin\theta \)
Therefore, a complex number in exponential form is given by:
\( z = r \cdot e^{i\theta} \)
Example 4: If \( z=5(\cos1.2 + i\sin1.2) \), then in exponential form:
\( z=5e^{i(1.2)} \)
Exponential form simplifies multiplication, division, and finding powers/roots.
5) De Moivre’s Theorem ✨
For \( z=r(\cos\theta+i\sin\theta) \), an integer power \( n \) is given by:
\( z^n = r^n\bigl(\cos(n\theta)+i\sin(n\theta)\bigr) \)
Example 5 (Compute Power):
Let \( z=2(\cos30^\circ+i\sin30^\circ) \). Then \( z^3=2^3\bigl(\cos90^\circ+i\sin90^\circ\bigr)=8(0+i)=8i \).
6) Complex Roots of Polynomials 🧩
6.1 nth Roots of a Complex Number
If \( z=r(\cos\theta+i\sin\theta) \), the n distinct nth roots are:
\( z_k=\sqrt[n]{r}\Bigl[\cos\Bigl(\frac{\theta+2k\pi}{n}\Bigr)+i\sin\Bigl(\frac{\theta+2k\pi}{n}\Bigr)\Bigr],\quad k=0,1,\dots,n-1 \)
Example 6 (Cube Roots):
Find all cube roots of \( 8(\cos45^\circ+i\sin45^\circ) \).
- \( r=8 \) so \( \sqrt[3]{8}=2 \).
- For \( k=0 \): Angle \( =\frac{45^\circ}{3}=15^\circ \). For \( k=1 \): Angle \( =\frac{45^\circ+360^\circ}{3}=135^\circ \). For \( k=2 \): Angle \( =\frac{45^\circ+720^\circ}{3}=255^\circ \).
Thus, the cube roots are:
- \( 2(\cos15^\circ+i\sin15^\circ) \)
- \( 2(\cos135^\circ+i\sin135^\circ) \)
- \( 2(\cos255^\circ+i\sin255^\circ) \)
6.2 Solving Polynomial Equations
For example, solve \( x^2+1=0 \). No real solutions exist, but in the complex domain:
\( x^2=-1 \Rightarrow x=\pm i \)
7) Examples (Detailed Solutions) 🍀
Example 8
Multiply in polar form:
Let \( z_1=3(\cos40^\circ+i\sin40^\circ) \) and \( z_2=2(\cos70^\circ+i\sin70^\circ) \).
\( z_1z_2=6\bigl(\cos(40^\circ+70^\circ)+i\sin(40^\circ+70^\circ)\bigr)=6(\cos110^\circ+i\sin110^\circ) \)
Example 9
Compute \( (1+i)^4 \) in polar form:
- Convert \( 1+i \) to polar form: \( |1+i|=\sqrt{2} \) and \( \theta=45^\circ \).
- Then, \( (1+i)^4=(\sqrt{2})^4\bigl(\cos(4\times45^\circ)+i\sin(4\times45^\circ)\bigr)=4\bigl(\cos180^\circ+i\sin180^\circ\bigr) \).
- Simplify: \( 4(-1+0i)=-4 \).
8) Practice Questions 🎯
8.1 Fundamental – Build Skills (10+)
- Convert \( z=3+4i \) to polar form \((r,\theta)\), then to exponential form \( re^{i\theta} \).
- Evaluate \( (2-3i)+(4+5i) \) and plot the result on the complex plane.
- Multiply \( (2+i)(1-2i) \) and simplify.
- Divide \( \frac{1+3i}{2-i} \) and express the result in \( a+bi \) form.
- Convert \( z=4(\cos30^\circ+i\sin30^\circ) \) to rectangular form \( a+bi \).
- Use De Moivre’s theorem to compute \( \Bigl(2(\cos45^\circ+i\sin45^\circ)\Bigr)^3 \).
- Solve \( x^2+1=0 \) and express the solutions in rectangular form.
- If \( z=5e^{i\pi/4} \), find \( z^2 \) using exponent rules and convert back to standard form.
- Find all 4th roots of \( 16(\cos0^\circ+i\sin0^\circ) \) and express each root in polar form.
- For the polynomial \( p(x)=x^2-(3+2i)x+(7+6i) \), if \( (x-(1+i)) \) is a factor, find the other factor.
8.2 Challenging – Push Limits (5+)
- 🔥 Solve \( |z-(2+3i)|=5 \) and find all points on this circle that satisfy \( \Re(z)=4 \).
- 🔥 Compute \( (1+i)^6 \) using polar form and compare with a direct binomial expansion.
- 🔥 Solve \( (x+yi)^3=-8 \) using De Moivre’s theorem and list all three complex solutions in polar form.
- 🔥 Find all cube roots of \( \sqrt{2}(\cos120^\circ+i\sin120^\circ) \) and express them in exponential form.
- 🔥 Solve \( \overline{z}=2z \) (where \( \overline{z} \) is the complex conjugate) and interpret the solutions.
9) Summary
- Advanced complex numbers use polar and exponential forms for easier multiplication, division, powers, and roots.
- De Moivre’s theorem states: \( \bigl(r(\cos\theta+i\sin\theta)\bigr)^n=r^n(\cos n\theta+i\sin n\theta) \).
- nth roots yield multiple solutions, evenly spaced on the complex plane.
- These concepts are fundamental in engineering, physics, and advanced mathematics.
Master these techniques to solve complex polynomial equations, compute powers and roots effortlessly, and tackle real-world applications requiring complex analysis! 🌟