Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
Complex Number Calculator
Enter complex numbers and choose an operation to see the result.
This calculator is also known as De Moivre's Theorem Calculator.
Read the complete guideWhat Is De Moivre's Theorem?
De Moivre's theorem says [r·cis(θ)]^n = r^n·cis(nθ). This means raising a complex number to a power is a simple operation in polar form: raise the modulus to the nth power and multiply the argument by n. The theorem also works for negative integers, giving 1/z^|n|.
Examples
(1 + i)^10 via De Moivre
Compute (1 + i) raised to the 10th power.
1 + i has modulus √2 and argument π/4. By De Moivre: (√2)^10 · cis(10π/4) = 32 · cis(5π/2) = 32 · cis(π/2) = 32i.
Key takeaway: De Moivre's theorem turns a tedious multiplication chain into a simple exponent and angle multiplication.
Applying De Moivre's Theorem
Steps for computing complex powers:
- Convert to polar form first: find r = |z| and θ = arg(z)
- Apply the formula: z^n = r^n · cis(nθ)
- Convert back to rectangular form if needed: a = r^n · cos(nθ), b = r^n · sin(nθ)
Frequently Asked Questions about De Moivre's Theorem Calculator
Does De Moivre's theorem work for negative exponents?
Yes. z^(−n) = 1/z^n = r^(−n) · cis(−nθ). The modulus becomes 1/r^n and the argument is negated.
What about z^0?
For any nonzero z, z^0 = 1 by convention. r^0 = 1 and cis(0) = 1.
Specialized Calculators
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