뒤로Powers of Complex Numbers (De Moivre’s Theorem)
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Powers of Complex Numbers
De Moivre’s Theorem
De Moivre’s Theorem is a fundamental result in trigonometry and complex numbers, allowing us to raise complex numbers in polar form to integer powers efficiently. This theorem is especially useful in simplifying powers and roots of complex numbers.
Statement of De Moivre’s Theorem: For any real number θ and integer n, if a complex number is written in polar form as z = r (cos θ + i sin θ), then:
Key Points:
Convert the complex number to polar form before applying the theorem.
Multiply the argument (angle) by n and raise the modulus (r) to the power n.
Worked Example
Given:
Find :
Evaluating the trigonometric values:
,
Generalization: The same process applies for higher powers, simply multiply the angle by n and raise the modulus to n.
Roots of Complex Numbers
To find the nth roots of a complex number, use the formula:
, where
Example: Find the square roots of
,
,
For : For :
Roots: and
Summary Table: De Moivre’s Theorem Applications
Operation | Formula | Description |
|---|---|---|
Power | Raise modulus to n, multiply angle by n | |
Root | Take nth root of modulus, divide angle plus by n |
Additional info: De Moivre’s Theorem is essential for simplifying complex number calculations in trigonometry, especially for powers and roots. It is also foundational for understanding the geometric interpretation of complex numbers on the Argand diagram.
