Euler and Roots Summary

Complex Analysis · Axiom Academy

SUMMARY Euler's Formula & Roots Unit 2 recap — the exponential view of complex numbers: Euler's formula, De Moivre's theorem, and the roots that fall out of them. Euler's formula rewrites a point on the unit circle as a single exponential — and gives the identity . Exponential form packs modulus and argument into one expression, making multiplication, powers, and roots almost mechanical. De Moivre's theorem : to raise to a power, raise the modulus and multiply the angle. Every nonzero number has exactly n n th roots, evenly spaced on a circle of radius — they trace a regular n -gon. The n th roots of unity sit on the unit circle and sum to 0 . The bridge between the exponential and the trigonometric. Feeding an imaginary exponent into e^ x traces the unit circle: the real part is , the imaginary part is . Identity: at it collapses to . Watch out for: is in radians , not degrees. Any complex number is its modulus r = |z| times , where is its argument. The same point as , written compactly. When to use: multiplying, dividing, powering, or taking roots. Watch out for: ; fix the quadrant when recovering . Core Concept De Moivre's Theorem Powers become arithmetic: raise the modulus to the n th power and multiply the angle by n . No binomial expansion needed. Rule of thumb: modulus to the power, angle times n . Works for: any integer n (and, read backward, fractional n for roots). For this lists all n roots of . They share modulus and are spaced apart — a regular n -gon.

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