Exponential Form

Complex Analysis · Axiom Academy

The most powerful notation for complex numbers — modulus and argument fused by Euler's formula. 1. One Arrow: Length r , Angle Picture z as an arrow from the origin. Its length is the modulus , and the angle it makes with the positive real axis is the argument . Euler's formula packages exactly those two numbers into . The length is r , the angle is Same number, three equal forms To write a number in exponential form you only need its length and its angle. Read the real part a and imaginary part b off the arrow, then: — the Pythagorean length of the arrow. Always positive. — the angle, chosen for the correct quadrant. In radians . . Done — the two pieces snap together. A bare misses half the plane; (or a quadrant check) fixes the angle. The arrow has length at , so . 3. Multiplying Adds the Angle Exponents Here is the payoff. Two numbers and multiply by the law of exponents: the lengths multiply and the angle exponents simply add . The product arrow is longer, and rotated by the second angle stacked on the first. Length ; angle . No FOIL, no i^2 bookkeeping — just multiply and add. 4. Powers Multiply the Angle by n Multiplying z by itself n times means adding its angle to itself n times. So raising to a power scales the length to r^n and the angle to — the dots march outward along a spiral, each step rotating by the same . This is De Moivre's theorem, and it makes roots just as easy in reverse.

This is the written version of the interactive lesson above. See the full Complex Analysis course.