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 a + bi and the root degree n.
2️⃣ The calculator converts the number into polar form:
r = √(a² + b²)
θ = atan2(b, a)
3️⃣ Using De Moivre’s Theorem, each root is calculated as:
zₖ = r^(1/n) · exp(i(θ + 2πk)/n)
for k = 0, 1, 2, ..., n−1
4️⃣ Each root is converted back to Cartesian form:
x = r^(1/n) cos((θ + 2πk)/n)
y = r^(1/n) sin((θ + 2πk)/n)
5️⃣ All roots are displayed in your selected format.
The roots are evenly spaced around a circle in the complex plane, separated by an angle of 2π/n.
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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