De Moivre's Theorem

This is an AoPSWiki Word of the Week for September 5- September 11

DeMoivre's Theorem is a very useful theorem in the mathematical fields of complex numbers. It allows complex numbers in polar form to be easily raised to certain powers. It states that for an $\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)$

Proof

This is one proof of De Moivre's theorem by induction.

If $n>0$ , for $n=1$ , the case is obviously true.

Assume true for the case $n=k$ . Now, the case of $n=k+1$ :

Therefore, the result is true for all positive integers $n$ .

If $n=0$ , the formula holds true because $\cos(0x)+i\sin (0x)=1+i0=1$ . Since $z^0=1$ , the equation holds true.

If $n<0$ , one must consider $n=-m$ when $m$ is a positive integer.

And thus, the formula proves true for all integral values of $n$ . $\Box$

By Euler's formula ($e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. Unknown error_msg)), this can be extended to all real numbers $n$ .

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De Moivre's Theorem

Proof