De Moivre's Theorem
Powers of complex numbers in polar form
About De Moivre's Theorem
The De Moivre's Theorem represents powers of complex numbers in polar form. This complex analysis formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.
This formula is essential in Complex analysis and Trigonometry. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.
Practical applications of the De Moivre's Theorem include Signal processing, Electrical engineering, Quantum mechanics, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.
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LaTeX Code
(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)
Formula Information
Difficulty Level
Prerequisites
Discovered
18th century
Discoverer
Abraham de Moivre
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Important Notes
De Moivre's theorem is a fundamental result for working with powers and roots of complex numbers. It follows directly from Euler's formula: (e^(iθ))ⁿ = e^(inθ). The theorem makes it easy to compute high powers of complex numbers and find all nth roots. It's essential for understanding roots of unity and periodic complex functions.
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Frequently Asked Questions
What is De Moivre's theorem?
De Moivre's theorem states that for any real number θ and integer n: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). This makes computing powers of complex numbers in polar form very easy. For example, to find (1+i)¹⁰, convert to polar form and apply the theorem.
How is this related to Euler's formula?
De Moivre's theorem follows directly from Euler's formula e^(iθ) = cos θ + i sin θ. Since (e^(iθ))ⁿ = e^(inθ), we get (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). Euler's formula is more general and works for any real exponent, while De Moivre's theorem is specifically for integer powers.
How do I use De Moivre's theorem to find roots?
To find all nth roots of a complex number z = r(cos θ + i sin θ), use: z^(1/n) = r^(1/n)[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)] for k = 0, 1, 2, ..., n-1. This gives n distinct roots equally spaced on a circle. For example, the cube roots of 1 are 1, e^(2πi/3), and e^(4πi/3).
What are roots of unity?
The nth roots of unity are the n solutions to zⁿ = 1. By De Moivre's theorem, they are: e^(2πik/n) for k = 0, 1, ..., n-1. These are equally spaced on the unit circle. For example, the 4th roots of unity are 1, i, -1, -i. They're fundamental in many areas of mathematics.
How is De Moivre's theorem used in practice?
De Moivre's theorem is used in: computing high powers of complex numbers (much easier than expanding), finding all roots of complex numbers, deriving trigonometric identities (like cos(3θ) in terms of cos θ), signal processing (phasor analysis), and solving polynomial equations with complex coefficients.
Can De Moivre's theorem be extended to non-integer exponents?
Yes! Using Euler's formula, (re^(iθ))^a = r^a e^(iaθ) works for any real exponent a. However, for non-integer a, you get multiple values (branch cuts). For integer n, De Moivre's theorem gives a unique result. For fractional powers, you get multiple roots as described in the root-finding formula.
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Quick Details
- Category
- Complex Analysis
- Difficulty
- Intermediate
- Discovered
- 18th century
- Discoverer
- Abraham de Moivre
- Formula ID
- demoivre