Prove that De Moivre's Theorem is true for all integers n by assuming it is true for integers n

step 1 When n equals 1, the statement says, if a is greater than 1, then a is greater than 1. And obvio

Prove that De Moivre's Theorem is true for all integers n by...

Key Concepts

Base Case and Extension to Zero

In the context of De Moivre's Theorem, verifying the base case when n = 0 is essential. Since any non-zero complex number raised to the power of zero equals 1, and noting that cos(0) + i sin(0) also equals 1, the theorem holds for n = 0. This case underpins the consistency of the theorem across different exponent values.

Extension to Negative Integers

Extending De Moivre's Theorem to negative integers involves recognizing that a negative exponent corresponds to the reciprocal of the complex number raised to the positive equivalent of that exponent. By using the property that (cos(?) + i sin(?))^-n equals 1/(cos(?) + i sin(?))^n, and applying the theorem for positive n, one confirms that the theorem remains valid for negative integer exponents.

De Moivre's Theorem

De Moivre's Theorem establishes that for a complex number in polar form, expressed as cos(?) + i sin(?), raising it to an integer power n results in cos(n?) + i sin(n?). This theorem is central in complex number theory as it links exponentiation with trigonometric functions, providing a straightforward method to compute powers and roots of complex numbers.

Mathematical Induction

Mathematical induction is a proof technique that verifies a statement for a base case and then assumes its truth for an arbitrary integer n to prove it for n+1. This approach is instrumental in establishing the validity of De Moivre's Theorem for all positive integers by systematically building from the known case to the general case.

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