Complex numbers: representation, arithmetic, modulus-argument form, loci, De Moivre’s theorem

Complex Numbers: A Journey into the Imaginary World 🌌

1. What is a Complex Number? 🤔

A complex number is a number that has two parts: a real part and an imaginary part. We write it as $a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ satisfies $i^2 = -1$. Think of it like a two‑dimensional dance: the real part moves along the horizontal axis, the imaginary part along the vertical axis.

2. Representing Complex Numbers: The Plane 📐

We can plot $a + bi$ on a graph called the complex plane. The horizontal axis (real axis) holds $a$, and the vertical axis (imaginary axis) holds $b$. The point $(a, b)$ is the location of the complex number.

3. Arithmetic with Complex Numbers ➕➖✖️➗

• Addition/Subtraction: Combine like parts. $$ (a+bi) \pm (c+di) = (a\pm c) + (b\pm d)i $$
• Multiplication: Use the distributive law and remember $i^2 = -1$. $$ (a+bi)(c+di) = (ac-bd) + (ad+bc)i $$
• Division: Multiply numerator and denominator by the conjugate of the denominator. $$ \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2} = \frac{(ac+bd)+(bc-ad)i}{c^2+d^2} $$

4. Modulus and Argument: Magnitude & Direction 📏🧭

The modulus (or absolute value) of $z = a+bi$ is its distance from the origin: $$ |z| = \sqrt{a^2 + b^2} $$ It tells us how far the point is from (0,0).

The argument (or angle) is the direction from the origin to the point, measured counter‑clockwise from the positive real axis: $$ \arg(z) = \tan^{-1}\!\left(\frac{b}{a}\right) $$ (adjusted for the correct quadrant).

Combining them gives the polar form: $$ z = |z|(\cos\theta + i\sin\theta) = |z|e^{i\theta} $$ where $\theta = \arg(z)$.

5. Loci of Complex Numbers: Where They Live 🏠

A locus is the set of all points that satisfy a given condition. Some common examples:

6. De Moivre’s Theorem: Powers & Roots in a Snap 🔁

For any integer $n$ and any complex number $z = r(\cos\theta + i\sin\theta)$, De Moivre’s theorem states:

This makes raising complex numbers to powers (and taking roots) very simple. For example:

• Compute $z^3$ for $z = 1 + i$:
  1. Find $r = \sqrt{1^2 + 1^2} = \sqrt{2}$ and $\theta = \tan^{-1}(1/1) = \pi/4$.
  2. Apply De Moivre: $$ z^3 = (\sqrt{2})^3 \bigl(\cos(3\pi/4) + i\sin(3\pi/4)\bigr) = 2\sqrt{2}\left(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) = -2 + 2i $$
• Find the cube roots of $8$:
  1. Write $8$ in polar form: $8 = 8(\cos 0 + i\sin 0)$.
  2. Apply De Moivre for $n = 3$: $$ \sqrt[3]{8} = 2\bigl(\cos(0/3 + 2k\pi/3) + i\sin(0/3 + 2k\pi/3)\bigr), \quad k = 0,1,2 $$
  3. So the roots are $2$, $-1 + i\sqrt{3}$, and $-1 - i\sqrt{3}$.

Remember: De Moivre’s theorem turns complex multiplication into simple angle addition and radius multiplication—just like adding angles when you rotate a spinning top!