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Comparing Moivre's Pattern Of The Equation With Complex Number

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Math IA Math Internal Assessment EF International Academy NY Student Name: Joo Hwan Kim Teacher: Ms. Gueye Date: March 16th 2012 Contents Introduction Part A Part B Conclusion Introduction The aim of this IA is to find out the pattern of the equations with complex numbers by using our knowledge. I used de Moivre’s theorem and binomial expansion, to find out the specific pattern and make conjecture about it. I basically used property of binominal theory with the relationship between the length of the line segments and the roots. Part A To obtain the solutions to the equation ) | | Moivre’s theorem, (| | equation, we will get: , I used de Moivre’s theorem. According to de . So if we apply this theorem in to the (| | ) …show more content…

And the distance of arc between each neighboring roots are same with others. Roots of this equation increase by are three roots on the unit circle. , so we can find that there When ( ( ( ) ) ) Suppose ( ) ( ( ( ) ) ) Figure 11 The graph of roots of equation z^4=i When n=5, ( ( ) ) Suppose ( ( ) ( ( ( ) ) ) ) ( ) Figure 12 The graph of roots of the equation z^5=i Basically all the roots we found are on the lane of the unit-circle, because we use the complex ( ) number whose modulus is 1. ] . So if I generalize the equation of , I would get: ( ( So for the equation like equation is Generalize the equations of , ) ) that satisfy this ( ). And I can should be (0+1i)= i. And the value of into , where n=3,4 and 5. rad. So we can change the equation ( ) ( ( ( ) ) ) With the knowledge of in the right triangle of a b So With the knowledge √ It is possible state that This generalization is proved naturally as we found out that the angle of the roots is . When But when | under the condition of | | | has a generalization of the generalization would change as Conclusion I found out some patterns about two different equation: some conjectures that led me to find out and prove it. For of all length of the line segments connected form a root to others. . There were n is equal to the

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