n-th Root of Complex Numbers

Calculation of \(\sqrt[n]{z}\) - n different solutions for each root

Root Calculator

n-th Root of Complex Numbers

The n-th root \(\sqrt[n]{z} = z^{1/n}\) of a complex number has n different solutions, which are evenly distributed on a circle. This calculator returns the principal value.

Base z = a + bi
+
i
Root exponent n (real number)
n
n=2: square root, n=3: cube root, n=4: fourth root, etc.
Calculation Result
\(\sqrt[n]{z}\) (principal value) =
There are n different solutions! This calculator shows only the principal value (k=0).

n-th Root - Properties

Formula in Polar Form
\[\sqrt[n]{z} = \sqrt[n]{r} \cdot e^{i(\phi + 2\pi k)/n}\]

With \(k = 0, 1, 2, ..., n-1\) (n different solutions)

Principal value (k=0)
\[w_0 = \sqrt[n]{r} \cdot e^{i\phi/n}\]
Magnitude \(\sqrt[n]{|z|}\)
Solutions n values
n different solutions!

The n-th root has n different solutions:
\[w_k = \sqrt[n]{r} \cdot e^{i(\phi + 2\pi k)/n}\] with \(k = 0, 1, 2, ..., n-1\)
These are evenly distributed on a circle!

Important Properties
  • \((\sqrt[n]{z})^n = z\) (definition)
  • \(|\sqrt[n]{z}| = \sqrt[n]{|z|}\) (magnitude)
  • \(\arg(\sqrt[n]{z}) = \frac{\arg(z)}{n}\) (angle divided by n)
  • Angular spacing: \(\frac{360°}{n}\) between solutions
Special Cases
  • n=2: square root (2 solutions)
  • n=3: cube root (3 solutions)
  • n=4: fourth root (4 solutions)
  • n→∞: infinitely many solutions

Formulas for the n-th Root of Complex Numbers

The n-th root of a complex number \(z\) is defined as \(\sqrt[n]{z} = z^{1/n}\) and has n different solutions.

Polar Form
For \(z = re^{i\phi}\): \[w_k = \sqrt[n]{r} \cdot e^{i(\phi + 2\pi k)/n}\]

With \(k = 0, 1, 2, ..., n-1\)

Geometric Distribution
Angular spacing: \(\frac{360°}{n}\)

The solutions lie evenly on a circle with radius \(\sqrt[n]{r}\)

Step-by-Step Example

Calculation: \(\sqrt[3]{8}\) (cube root of 8)
Step 1: Polar form

z = 8 + 0i

r = |z| = 8

φ = arg(z) = 0°

Step 2: Principal value (k=0)

\(w_0 = \sqrt[3]{8} \cdot e^{i \cdot 0/3}\)

\(= 2 \cdot e^{i \cdot 0}\)

\(= 2\)

Step 3: Additional solutions

k=1: \(w_1 = 2e^{i \cdot 120°}\)

\(= 2(\cos 120° + i\sin 120°)\)

\(\approx -1 + 1.732i\)

k=2: \(w_2 = 2e^{i \cdot 240°}\)

\(= 2(\cos 240° + i\sin 240°)\)

\(\approx -1 - 1.732i\)

All 3 solutions:
\(w_0 = 2\)
Angle: 0°
\(w_1 \approx -1 + 1.732i\)
Angle: 120°
\(w_2 \approx -1 - 1.732i\)
Angle: 240°
Verification

\(w_0^3 = 2^3 = 8\) ✓

\(w_1^3 = (-1+1.732i)^3 \approx 8\) ✓

\(w_2^3 = (-1-1.732i)^3 \approx 8\) ✓

More Examples

Example 1: \(\sqrt[4]{1}\) (4 roots of unity)

r = 1, φ = 0°

Angular spacing: 360°/4 = 90°

\(w_0 = 1\) (0°)
\(w_1 = i\) (90°)
\(w_2 = -1\) (180°)
\(w_3 = -i\) (270°)

Example 2: \(\sqrt[3]{-8}\)

r = 8, φ = 180°

Angular spacing: 360°/3 = 120°

\(w_0 = 2e^{i60°} \approx 1 + 1.732i\)
\(w_1 = 2e^{i180°} = -2\)
\(w_2 = 2e^{i300°} \approx 1 - 1.732i\)

Example 3: \(\sqrt[2]{i}\)

r = 1, φ = 90°

\(w_0 = e^{i45°}\)
\(\approx 0.707 + 0.707i\)
\(w_1 \approx -0.707 - 0.707i\)

Example 4: \(\sqrt[3]{1+i}\)

r = √2, φ = 45°

\(w_0 = \sqrt[3]{\sqrt{2}}e^{i15°}\)
\(\approx 1.084 + 0.290i\)
(2 more solutions)

Example 5: \(\sqrt[5]{32}\)

r = 32, φ = 0°

5 solutions with
angular spacing 72°
Principal value: \(w_0 = 2\)

n-th Root of Complex Numbers - Detailed Description

Definition and Multivaluedness

The n-th root \(\sqrt[n]{z}\) is defined as the solution of the equation \(w^n = z\).

Multivaluedness:
Every complex number \(z \neq 0\) has n different n-th roots:
\[w_k = \sqrt[n]{r} \cdot e^{i(\phi + 2\pi k)/n}\]
with \(k = 0, 1, 2, ..., n-1\)

Principal value:
The principal value is \(w_0\) (for \(k=0\))

Geometric Distribution

The n solutions are particularly beautifully distributed:

Uniform distribution:

  • All lie on a circle with radius \(\sqrt[n]{r}\)
  • Angular spacing: \(\frac{360°}{n} = \frac{2\pi}{n}\)
  • Form a regular n-gon
  • Symmetric about the origin

Calculation

In polar form, the calculation is especially simple:

For \(z = re^{i\phi}\):
\[w_k = \sqrt[n]{r} \cdot e^{i(\phi + 2\pi k)/n}\]
Steps:
1. Magnitude: \(\sqrt[n]{r}\) (take n-th root)
2. Angle: \(\frac{\phi + 2\pi k}{n}\) (divide by n, +2πk/n)
3. Calculate for all k = 0, 1, ..., n-1

Important Properties

  • \((\sqrt[n]{z})^n = z\) (for all n solutions)
  • \(|\sqrt[n]{z}| = \sqrt[n]{|z|}\) (magnitude)
  • \(\arg(w_k) = \frac{\arg(z) + 2\pi k}{n}\) (argument)
  • \(\sqrt[n]{z_1 \cdot z_2} \neq \sqrt[n]{z_1} \cdot \sqrt[n]{z_2}\) (caution!)
  • Roots of unity: \(\sqrt[n]{1}\) form cyclic group

Roots of Unity

The n-th roots of unity \(\sqrt[n]{1}\):

\[\omega_k = e^{i \cdot 2\pi k/n}\]

With \(k = 0, 1, ..., n-1\)
Important in algebra and Fourier transformation

Practical Applications

Mathematics:
• Algebra: solving polynomial equations
• Number theory: roots of unity
• Galois theory: field extensions
• FFT: Fast Fourier Transform
Applications:
• Signal processing: filter design
• Cryptography: primitive roots
• Quantum mechanics: phases
• Fractals: Julia sets
Tip: Visualization

Plot the n solutions in the complex plane:
They form a regular n-gon on a circle!
This helps understand the structure of the solutions.


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Airy function • Derivative Airy function •
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