Complex Numbers

The Key Result

Why It Works

1. 1
   Define $i$ by $i^2 = -1$; then $z = a + bi$ is a point $(a, b)$ in the plane with modulus $|z| = \sqrt{a^2 + b^2}$ and argument $\theta = \operatorname{atan2}(b, a)$.
2. 2
   Write $z = r(\cos\theta + i\sin\theta)$ with $r = |z|$; Euler's formula gives the compact $z = re^{i\theta}$.
3. 3
   Multiplying $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$ gives $r_1 r_2 e^{i(\theta_1 + \theta_2)}$ — moduli multiply, arguments add.
4. 4
   De Moivre's theorem follows: $z^n = r^n\big(\cos n\theta + i\sin n\theta\big)$, turning powers into angle multiplication.

Worked Examples

Going Deeper

• Generalization: identifying $\mathbb{C}$ with the plane turns multiplication by $re^{i\theta}$ into 'scale by $r$, rotate by $\theta$'; this is why complex numbers solve plane-geometry rotation problems and why every degree-$n$ polynomial has exactly $n$ roots (Fundamental Theorem of Algebra).
• Where it appears: AMC/AIME problems on powers of $1 \pm i$, sums like $\sum i^k$, rotating a point about another in the plane (multiply the difference by $e^{i\theta}$), and as the engine behind De Moivre's derivations of multiple-angle trig identities.
• Pitfall: $\arg$ is only determined up to multiples of $2\pi$, and $\sqrt{\;}$ / $\theta = \arctan\frac{b}{a}$ pick the WRONG branch when the point is not in the first quadrant — always check which quadrant $(a, b)$ lies in (use $\operatorname{atan2}$) before assigning the argument, or De Moivre will rotate you to the wrong answer.

Spot the Signal

• Look for problems where the key step is using real and imaginary parts, modulus, argument, and roots to encode algebraic structure.
• You can describe the hard part as using real and imaginary parts, modulus, argument, and roots to encode algebraic structure, but a direct attack starts producing clutter.
• The problem rewards preserving structure instead of expanding, listing, or guessing too early.

Learn the Move

• Start by identify the expression or equation that calls for complex numbers, then rewrite the givens around it.
• Name the relation that makes Complex Numbers legal before doing computation.
• Use the new relation to replace the messiest part of the problem with a cleaner one.
• Translate the result back to the quantity the problem actually asks for.

Avoid These Traps

• Do not use Complex Numbers just because the surface looks familiar; verify the required condition first.
• Applying Complex Numbers because it sounds relevant, without checking the trigger first.
• Stopping after spotting the technique instead of finishing the calculation or proof.

MathGrit Coach Note