Find all n-th roots using De Moivre’s Theorem.
This calculator finds all n-th roots of a complex number a + bi using De Moivre’s Theorem. It converts the number into polar form, computes every possible root, and displays them in Cartesian form, Polar form, or both — instantly.
Enter:
The calculator computes all n distinct complex roots.
Choose how results are displayed:
Perfect for algebra, engineering, and complex analysis.
Enable precision control to:
Useful for academic and professional calculations.
The calculator:
1️⃣ Enter the complex number.
Start by typing the real part (a) and imaginary part (b) of the complex number.
Together these values represent the number in the form a + bi.
2️⃣ Choose the root degree.
Next, enter the value of n, which represents the degree of the root you want to calculate.
For example, entering 2 will calculate square roots, while entering 3 will calculate cube roots.
3️⃣ The calculator converts the number to polar representation.
Instead of working directly with the standard form, the calculator internally transforms the complex number
into a polar representation that describes the number using its magnitude and angle in the complex plane.
This representation makes root calculations much easier.
4️⃣ All possible roots are generated.
Using De Moivre’s Theorem, the calculator determines every valid root of the complex number.
A complex number always has exactly n distinct n-th roots, and each root corresponds to a different
angle around the complex plane.
5️⃣ The roots are converted into readable form.
After the roots are computed, the calculator converts them into the format you selected:
6️⃣ The results are displayed instantly.
Each root is shown in a clear card format. If precision control is enabled, the values are rounded
to the number of decimal places you selected.
7️⃣ Input validation ensures accurate results.
If any field is missing or the root degree is invalid, the calculator automatically hides the result
and shows an error message until valid values are entered.
Because the roots are evenly distributed around a circle in the complex plane, the results often form symmetrical geometric patterns. This property is widely used in signal processing, electrical engineering, and advanced mathematics.
Finding the n-th roots of a complex number is based on an important concept in complex analysis known as De Moivre’s Theorem. Instead of working directly with the Cartesian form a + bi, the number is first expressed in polar form, which represents the magnitude and angle of the number in the complex plane. Using polar form makes it possible to compute multiple roots easily and understand their geometric structure.
zₖ = r^(1/n) · exp(i(θ + 2πk)/n)
In this formula:
Each value of k produces a different solution, which means a complex number has exactly n distinct n-th roots. These roots are evenly spaced around a circle in the complex plane, forming a symmetric pattern. This geometric structure is one of the most interesting properties of complex numbers and is widely used in mathematics, physics, signal processing, and engineering.
For example, when calculating the square roots of a complex number, the two solutions appear opposite each other on the complex plane. For higher roots such as cube roots or fourth roots, the solutions form equally spaced points around the circle.
Suppose:
z = 1 + i, n = 2
The calculator computes two square roots:
√(1 + i) → Two complex roots
These roots are positioned opposite each other on the complex plane (180° apart), demonstrating how n-th roots form symmetric patterns.
Everything you need to know about finding all n-th roots of a complex number using De Moivre’s Theorem.
This calculator finds all n-th roots of a complex number a + bi using De Moivre’s Theorem. It converts the number to polar form, applies the root formula, and displays all distinct roots.
You must enter the real part (a), imaginary part (b), and the root degree (n). The degree must be a positive integer.
The calculator will display an error message. The root degree must be greater than zero because only positive integer roots are mathematically valid in this context.
The calculator first converts a + bi into polar form r·exp(iθ). Then it computes each root using the formula: r^(1/n) · exp(i(θ + 2πk)/n) for k = 0 to n − 1. This ensures all distinct roots are generated.
The calculator always displays exactly n distinct roots, where n is the root degree you enter.
You can choose Cartesian form (a + bi), Polar exponential form (r · exp(iφ)), or Both forms together. The calculator displays results based on your selection.
If you enable “Set Output Precision,” you can choose the number of decimal places (0 to 15). All real and imaginary values will be rounded to that precision.
Yes. If b = 0 (purely real) or a = 0 (purely imaginary), the calculator still converts the number to polar form correctly and computes all roots.
If any required input (real part, imaginary part, or degree) is missing or invalid, the calculator hides the results and waits for valid values.
Yes. Use the “Clear all” button to reset inputs and results, or click “Reload calculator” to fully restore the default state.
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