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
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
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
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