Proving De Moivre's Theorem Using Mathematical Induction

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Proving De Moivre’s theorem using mathematical induction
000416 - 0010
Luis Blanco Tejada
Mathematics Standard Level
2nd of October of 2015

Introduction

When I first encountered De Moivre’s theorem I was quite skeptical with my math teacher, as it seemed too easy, difficult to believe blindly. To solve my doubts I will use this exploration as its aim is to proof by induction De Moivre’s theorem for all integers; using mathematical induction.
De Moivre was a French mathematician exiled in England, famous for his mathematical developments relating complex numbers to trigonometry. Between his acquaintances we can find Newton, Edmond Halley and other important characters in mathematics and science through History. His theorem or Formula
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The theorem

De Moivre’s theorem first appeared in his work as:
(cos⁡x+i sin⁡x )^n=1/2 (cos⁡nx+i sin⁡nx )+1/2 (cos⁡nx+i sin⁡nx ) Equation 1.1
This was later simplified in the form that is known nowadays as De Moivre’s theorem:
(r(cos⁡θ+i sin⁡θ))^n=r^n (cos⁡〖(nθ〗)+i sin⁡(nθ)) Equation 1.2
Where i is the imaginary number unit (i^2=-1)
Sometimes it is also common to abbreviate it in the form: CiS θ Equation 1.3. However this is just a simple abbreviation being the correct form is equation 1.2.
As an example of the usefulness of this theorem, when we multiply or divide complex numbers we do it when possible in polar form because of the easiness of this using De Moivre’s theorem. An example of this is shown below:
If we want to multiply Z1 times Z2 when Z1 = 1+i√3 and Z2 = √2-i√2.
We encounter the problem that complex numbers in Cartesian form cannot be multiplied easily. Therefore we would want to convert them to polar form and then multiply
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It might not be Pythagoras theorem or Newton’s laws of motion. Nevertheless, this theorem is applied to complex numbers mostly whenever they are used. It may appear that complex numbers do not have a very straight application in real life mathematics; however, they can always be used to model a phenomenon. If this phenomenon is periodic and it is modeled with a complex number, De Moivre’s theorem would provide the solution to change the frequency of our model. In other words, this theorem would be the key to solve an ultimate issue with models, as we would use the exponent to do so. As an example, If we want to express repetitions of rotation in two planes (x and y) “n” times, we would use the theorem to raise the power n times and therefore alter the frequency of rotation. This is commonly used in

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