Find all nth roots of a complex number using De Moivre's theorem. Essential for advanced algebra, signal processing, and engineering.
Last updated: 2026-05-24T22:58:31.507Z | By ForgeCalc Engineering
Provide real and imaginary parts and an integer root degree (1–10).
1 to 10 roots
Complex Number (z)
1.0000
Magnitude (r):
1.0000
Angle (θ):
0.00°
The nth root of a complex number z is any complex number w such that w^n = z. Every non-zero complex number has exactly n distinct nth roots. This symmetry property is one of the beautiful characteristics of complex numbers.
The n roots are evenly spaced around a circle in the complex plane, separated by equal angles of 360°/n. This creates a perfectly symmetric pattern that reveals deep mathematical structure. All roots have the same distance from the origin.
If z = r·(cos θ + i sin θ), then the nth roots are:
wk = ⁿ√r · [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
where k = 0, 1, 2, ..., n−1
Every non-zero complex number has exactly n distinct nth roots
All roots have the same magnitude ⁿ√r
Each root differs by 360°/n from adjacent roots
Creates perfect rotational symmetry
|w_k| = ⁿ√|z| for all k
All roots are equidistant from origin
w₀ = ⁿ√r · [cos(θ/n) + i sin(θ/n)]
Generated when k = 0
w_k = w₀ · e^(i·2πk/n) = w₀ · ζ^k
Where ζ is a primitive nth root of unity
w₀ · w₁ · ... · w_{n-1} = z for certain products
Roots relate back to original number
Every non-zero complex number has exactly n distinct nth roots. For example, every number (except 0) has 2 square roots, 3 cube roots, 4 fourth roots, and so on.
The n-th roots of a complex number are generated by the formula w_k = w₀·e^(i·2πk/n). The exponential factor e^(i·2πk/n) rotates by equal angles, creating circular symmetry.
The three cube roots of 1 are: 1, e^(i·2π/3) = -1/2 + i√3/2, and e^(i·4π/3) = -1/2 - i√3/2. These are called the primitive cube roots of unity.
The principal nth root corresponds to k=0 in the formula. Convert z to polar form, take the nth root of the magnitude, and divide the angle by n.
De Moivre's Theorem states that (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). This powerful tool connects algebra and trigonometry, enabling nth root calculations.
Complex roots are essential in AC circuit analysis, control systems, signal processing, and stability analysis. They help solve differential equations that model physical systems.
Polar Form Conversion
z = r(cos θ + i sin θ) = r·e^(iθ)
where r = √(a² + b²) and θ = atan2(b, a)
General nth Root Formula
w_k = ⁿ√r · e^(i(θ + 2πk)/n)
for k = 0, 1, 2, ..., n−1
Angular Separation
Δθ = 360°/n
Angle between consecutive roots
Related Tools
Solve absolute value equations.
Solve absolute value inequalities.
Apply associative property.
Multiply using box method.
Complete the square method.
Solve using Cramer's Rule.