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You are here: Home / Category Index / Complex Numbers / De Moivre's Theorem
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De Moivre's Theorem - By Thomas Deloford
How to use this applet:Follow the steps 1 - 4 as described on the right of the applet.
Use of the Graph: Use the right button while over the graph to commit your selected complex to raise, then enter a real positve value to raise your complex by. To reset and perform another calculation simply "right click" over the graph, this will allow you to modify your previous complex.Notes on the maths used in the applet:Proof by Induction:
De Moivre's Theorem can be proved for all real numbers as follows;
Assuming the theorem to be true for n = k
(CosX + i SinX) ^ k = Cos(k*X) + i Sin(k*X)
Now Consider n = k + 1:
(Cos X + i Sin X)^k+1 = (Cos X + i Sin X) ^ k * (Cos X + i Sin X)
= (Cos(k*X) + i Sin(k*X)) ^ k * (Cos X + i Sin X)
= Cos(k*X + X) + i Sin(k*X +X)
= Cos(k + 1) * X + i Sin(k + 1) * X
So if the theorem is true for n = k, it is also true for n = k + 1.
when n = 1 (Cos X + i Sin X) ^ 1 = Cos X + i Sin X
= Cos(1*X) + i Sin(1*X)
The theorem is true for n = 1.
Hence by induction De Moivre's theorem is true when n is a positive integer
Other useful information: Before you use this applet you must first know what a complex number is, understand the Modulus argument and how to derive this from a complex in the form a + bi (see Modulus Argument). It is also advisable to have an understanding of division of complex numbers, because the proof by Induction assumes this Knowledge.
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