Complex numbers
Lab: all complex roots
Solve zⁿ=w, construct every root and check one solution at a time. Change the argument without changing the number: the points stay the same.
Learning-path index · Read the article · Stage 5: Euler and De Moivre
The problem you are solving
Inputs and general formula
Convention: Arg(w)∈(−π,π]. The principal root uses Arg(w)/n; it need not have k=0 when turns are added. All other roots remain valid solutions.
Select a point or row. On the plot: left/right arrows, Home/End; Enter selects the focused point. Animation highlights one root per step, not intermediate solutions.
All roots in the plane
Every point is a solution; orange = selected, purple ring = principal. Edges are geometric guides, not additional solutions. The two plots use independent scales shown by tick labels.
Selected root
Verification by taking the power
The green point w and orange result z_kⁿ coincide mathematically. Numerical calculations may leave a small rounding residual.
Coordinates marked ≈ and the residual use floating-point numbers. Even a displayed residual of 0 is not by itself a certificate of exact equality; the formula and rational checks are separate from the numerical test.
Table of all roots
Select k to choose a root. On mobile the table scrolls horizontally. Angles are unwrapped: added turns change labels, not points.
| k | Angle θ_k | Coordinates | Principal root |
|---|