MathematicsComplex Numbers

De Moivre

De Moivre's theorem provides a direct method for calculating powers of complex numbers expressed in polar form. It demonstrates that raising a complex number to an integer power n is equivalent to multiplying its argument by n while the modulus is raised to that power.

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(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)

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Core idea

Overview

When to use: This theorem is essential when raising complex numbers to high integer powers or when solving for the n-th roots of unity. It requires the complex number to be in polar or trigonometric form before the exponent is applied.

Why it matters: It serves as a vital link between trigonometry and complex algebra, simplifying calculations in electrical engineering, signal processing, and fluid dynamics. By converting multiplication into addition of angles, it bypasses the need for complex binomial expansions.

Remember it

Memory Aid

Phrase: Cousins In Sync: Power n slides inside.

Visual Analogy: Think of a rotating dial. Turning it by an angle theta exactly n times is the same as one single large rotation of n-theta.

Exam Tip: Only use this when the real part is Cos and the imaginary is Sin. If they are swapped, use trig identities to fix the format first.

Why it makes sense

Intuition

De Moivre's theorem geometrically describes how raising a complex number on the unit circle to an integer power n causes its angular position to rotate n times its original angle around the origin, while remaining on the

Symbols

Variables

$θ$ = Angle Î¸, n = Power n

θ

Angle Î¸

deg

n

Power n

Variable

Walkthrough

Derivation

Understanding De Moivre's Theorem

De Moivre’s theorem raises complex numbers in polar form to integer powers by raising the modulus and multiplying the argument.

z is in polar form $z = r (cos θ + i sin θ)$ .
n is an integer.

State the Theorem:

The modulus becomes $r^{n}$ and the argument becomes $n θ$ .

[r (cos θ + i sin θ)]^{n} = r^{n} (cos (n θ) + i sin (n θ))

Note: A standard proof uses mathematical induction.

Result

[r (cos θ + i sin θ)]^{n} = r^{n} (cos (n θ) + i sin (n θ))

Source: OCR Further Mathematics — Core Pure (Complex Numbers)

Where it shows up

Real-World Context

Signal processing phases.

Avoid these traps

Common Mistakes

Applying to non-polar form.
Forgetting magnitude power.

Study smarter

Tips

Always convert Cartesian coordinates (a + bi) to polar form (r, Î¸) first.
Remember that the exponent n multiplies the angle but acts as a power for the modulus r.
Check that the angle is consistently in either degrees or radians before calculating.
The formula remains valid for negative integers, which corresponds to a change in rotation direction.

Common questions

Frequently Asked Questions

De Moivre’s theorem raises complex numbers in polar form to integer powers by raising the modulus and multiplying the argument.

This theorem is essential when raising complex numbers to high integer powers or when solving for the n-th roots of unity. It requires the complex number to be in polar or trigonometric form before the exponent is applied.

It serves as a vital link between trigonometry and complex algebra, simplifying calculations in electrical engineering, signal processing, and fluid dynamics. By converting multiplication into addition of angles, it bypasses the need for complex binomial expansions.

Applying to non-polar form. Forgetting magnitude power.

Signal processing phases.

Always convert Cartesian coordinates (a + bi) to polar form (r, Î¸) first. Remember that the exponent n multiplies the angle but acts as a power for the modulus r. Check that the angle is consistently in either degrees or radians before calculating. The formula remains valid for negative integers, which corresponds to a change in rotation direction.