Complex numbers

Lab: Euler, De Moivre and powers

Explore the unit circle, integer powers and finite sums of Euler’s series. Every step distinguishes exact values, when available, from drawings and numerical approximations.

Path index · Read stage 5 · Review trigonometric form

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Angle θAngle unit

Coordinates −20 to 20 and modulus 0 to 20: integers, decimals up to three places or fractions of integers up to six digits. Angles −360° to 360° or π coefficients −2 to 2. n is an integer from −24 to 24; M is an integer from 0 to 40. Changing units converts the angle. Irrelevant fields do not enter the calculation.

Problem statement

Speed

Progress

On the circle the angle changes continuously. In the other modes Next advances one integer index; at normal speed animation advances one index per second.

Circle, powers and approximations in the plane

P is the selected point; E is e^(iθ), the reference in Euler and series modes. Lines connect points as visual guides, not fractional exponents.

A common automatic scale is used for all steps; widely different moduli may overlap visually. Read the numerical values. Drawings and generic trigonometric coordinates are approximate; Cartesian powers with fractions are calculated exactly.

Lab: Euler, De Moivre and powers

Problem statement

Circle, powers and approximations in the plane

Requested final result

Selected step

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