What does this calculator do?

This calculator solves w^n = z for a complex number z. In the complex plane, an nth-root problem has n roots. The tool is designed to show the full set, not just a single selected root.

• Use rectangular input for z = a + bi.
• Use polar input when you already know the radius r and angle θ in degrees.
• Choose an integer n from 2 to 20.
• Review every root in polar and rectangular form.
• Use the circle plot to understand the spacing between the roots.

How are nth roots of a complex number calculated?

Complex roots are easiest to understand in polar form. If z = r cis(θ), each nth root has radius r^(1/n), and its angle is (θ + 360°·k) / n for k = 0, 1, 2, ..., n−1.

1. Convert z to radius and angle if rectangular input is used.
2. Compute the root radius r^(1/n).
3. Add 360°·k to the angle for each k.
4. Divide each adjusted angle by n.
5. Convert every polar root back to rectangular form.

Rectangular input vs polar input

Rectangular form separates the real and imaginary parts as a+bi. Polar form describes the same number by distance from the origin and angle. The calculation itself follows the polar-root pattern, so polar input is convenient when the angle is already known.

• Rectangular input: enter a and b, and the calculator derives radius and angle.
• Polar input: enter r and θ directly; r cannot be negative.
• Both input modes represent the same complex number when they describe the same point.
• Angles are entered in degrees, not radians.

Example: fourth roots of 16

For z = 16 + 0i and n = 4, the root radius is 16^(1/4) = 2. The four angles are 0°, 90°, 180°, and 270°, so the roots are 2, 2i, −2, and −2i.

1. Enter z as 16 + 0i.
2. Set n to 4.
3. The calculator creates roots for k=0,1,2,3.
4. All roots have the same radius.
5. The rectangular outputs appear approximately as 2+0i, 0+2i, −2+0i, and 0−2i.

Principal root vs all complex roots

The principal root is one selected root, often the one corresponding to the principal angle. But the equation w^n = z has n complex solutions. This calculator is built for the complete solution set.

• A principal-root function may return only one value.
• The full nth-root set uses all k values from 0 to n−1.
• For equation solving, using only the principal root can omit valid solutions.
• The circle plot helps reveal the symmetry between the roots.

Limits and correct use

This is a numeric complex-root calculator. It does not try to become a general symbolic equation solver or a proof tool.

• n must be an integer from 2 to 20.
• The n≤20 cap keeps the result table and visual plot readable and follows the project’s existing root-index convention.
• Polar input must use a nonnegative radius.
• Results are numerical approximations.
• The circle plot is explanatory, not a precision graphing environment.