Powers, roots and circles in the Argand diagram
Use e^{iθ} as a point on the unit circle, raise numbers to powers with De Moivre, find the four roots of a number as a square and draw a circle with |z − c| = r.
- UK: Year 12
- France: grade 12
- Syria: grade 12
- Russia: grade 11
Guide
The number eiθ is the point on the unit circle at angle θ: eiθ = cosθ + isinθ. A number with modulus r and argument θ is reiθ. Half a turn is π radians, so eiπ = −1.
De Moivre: to raise a number to the power n, raise its modulus to the power n and multiply its argument by n. The n roots of a number all have the same modulus and their arguments are 360° ÷ n apart, so they are the corners of a regular polygon. The roots of zn = 1 are the roots of unity, on the unit circle.
The distance from z to a fixed point c is |z − c|. So |z − c| = r is a circle with centre c and radius r.