de moivre's theorem
Very lowTechnical/academic
Definition
Meaning
A formula in complex numbers that connects complex numbers and trigonometry, stating that for any real number x and integer n, (cos x + i sin x)^n = cos(nx) + i sin(nx).
A fundamental theorem in complex analysis and trigonometry used to compute powers and roots of complex numbers, with applications in electrical engineering, quantum mechanics, and signal processing.
Linguistics
Semantic Notes
Always capitalized (De Moivre's). Refers specifically to Abraham de Moivre. In mathematics, it's a proper noun naming a theorem.
Dialectal Variation
British vs American Usage
Differences
No significant differences in usage; both use the same term identically in mathematical contexts.
Connotations
Purely technical with no regional connotations.
Frequency
Equally rare in both varieties, appearing only in advanced mathematics education and research.
Vocabulary
Collocations
Grammar
Valency Patterns
[Subject] uses/applies/states/proves De Moivre's theorem to [mathematical object]Vocabulary
Synonyms
Neutral
Weak
Usage
Context Usage
Business
Never used.
Academic
Exclusively used in advanced mathematics, engineering, and physics courses and publications.
Everyday
Never used.
Technical
Used in mathematical proofs, engineering calculations involving complex numbers, and scientific research.
Examples
By CEFR Level
- De Moivre's theorem helps simplify calculations with complex numbers.
- We learned about De Moivre's theorem in our maths class.
- By applying De Moivre's theorem, one can easily compute the fifth power of a complex number in polar form.
- The proof of De Moivre's theorem for integer exponents relies on mathematical induction.
Learning
Memory Aids
Mnemonic
De Moivre moves the angle: cos + i sin to the power n becomes cos(nx) + i sin(nx).
Conceptual Metaphor
MATHEMATICAL RELATIONSHIP IS A BRIDGE (connecting powers to angles).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating 'theorem' as 'теория' (theory); correct is 'теорема'.
- Do not omit the possessive 's' in translation; it's 'теорема Муавра'.
- Do not confuse with Euler's formula, which is related but different.
Common Mistakes
- Misspelling as 'De Moiver's theorem' or 'De Moivres theorem'.
- Forgetting to capitalize 'De'.
- Using it with non-integer n without proper extension.
- Confusing it with Euler's formula e^(ix) = cos x + i sin x.
Practice
Quiz
What does De Moivre's theorem primarily relate?
FAQ
Frequently Asked Questions
Abraham de Moivre was an 18th-century French mathematician known for his work in probability theory and complex numbers.
The original theorem is for integer n, but it can be extended to rational and real numbers with careful interpretation.
It's used in electrical engineering to analyze AC circuits and in signal processing for Fourier transforms.
Euler's formula e^(ix) = cos x + i sin x is more general; De Moivre's theorem (cos x + i sin x)^n = cos(nx) + i sin(nx) can be derived from it for integer n.