Complex Number Input
De Moivre's Result
De Moivre's Theorem Calculator is an advanced mathematical tool used to quickly calculate the power of complex numbers. Instead of manually multiplying complex numbers in their standard form (a + bi), this theorem converts them into polar form to make exponential calculations simple and highly accurate.
How De Moivre's Theorem Works
The theorem states that for any complex number given in polar form z = r(cos θ + i sin θ), raising it to the power of n yields:
zⁿ = rⁿ(cos nθ + i sin nθ)
This means you simply raise the magnitude (r) to the power of n, and multiply the angle (θ) by n. The calculator then converts this final polar result back into standard rectangular form (a + bi) for easy reading.
How to Use This Mathematical Tool
- Enter the Real Part (a) of your base complex number.
- Enter the Imaginary Part (b) of your base complex number.
- Input the Power or Exponent (n) you want to raise the number to.
- The calculator will automatically process the polar conversion and apply De Moivre's theorem.
- Review your final complex number and the step-by-step magnitude and angle conversions in the result grid.
Frequently Asked Questions
Why is De Moivre's Theorem useful?
Expanding expressions like (1 + i) to the power of 10 manually is incredibly tedious and prone to algebraic errors. De Moivre's theorem bypasses the need for massive polynomial expansion by utilizing trigonometric geometry, providing the exact answer in a fraction of the steps.
Can the power (n) be a negative number or fraction?
Yes. De Moivre's theorem holds true for negative integers, which finds the inverse powers. It also applies to fractional powers (like 1/2 or 1/3), which is how mathematicians find the complex roots of numbers (such as finding the cube roots of unity).