Difference between revisions of "De Moivre's Theorem"

Revision as of 20:10, 23 November 2007

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 $n$ .

@@ Line 1: / Line 1: @@
-DeMoivre's Theorem is a very useful theorem in the mathematical fields of [[Complex numbers]]. It states that:
+'''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It allows complex numbers in [[polar coordinates|polar]] form to be easily raised to certain powers. It states that for an <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>
-<math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>
 == Proof ==
-This is one proof of DeMoivre's theorem by Mathematical Induction.
+This is one proof of De Moivre's theorem by [[induction]].
-=== If <math>n>0</math> ===
+*If <math>n>0</math>, for <math>n=1</math>, the case is obviously true.
-==== Part 1 ====
-For <math>n=1</math>, the case is obviously true.
-==== Part 2 ====
+:Assume true for the case <math>n=k</math>. Now, the case of <math>n=k+1</math>:
-Assume true for the case <math>n=k</math>.
-==== Part 3 ====
+:[[Image:DeMoivreInductionP1.gif]]
-Now, the case of <math>n=k+1</math>.
-[[Image:DeMoivreInductionP1.gif]]
+:Therefore, the result is true for all positive integers <math>n</math>.
-Therefore, the result is true for all positive integers <math>n</math>.
+*If <math>n=0</math>, the formula holds true because <math>\cos(0x)+i\sin (0x)=1+i0=1</math>. Since <math>z^0=1</math>, the equation holds true.
-=== If <math>n=0</math> ===
+*If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer.
-The formula holds true when <math>n=0</math> because <math>\cos(0x)+i\sin (0x)=1+i0=1</math>. Since <math>z^0=1</math>, the equation holds true.
-=== If <math>n<0</math> ===
+:[[Image:DeMoivreInductionP2.gif]]
-If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer.
-Therefore:
+And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math>
-[[Image:DeMoivreInductionP2.gif]]
+By [[Euler's formula]] (<math>e^{ix} = \cos x+i\sin x\right</math>), this can be extended to all real numbers <math>n</math>.
-And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math>
+[[Category:Theorems]]

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Difference between revisions of "De Moivre's Theorem"

Revision as of 20:10, 23 November 2007

Proof