Math Portfolio Type 2 - PATTERNS FROM COMPLEX NUMBERS

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Erzurum İhsan Doğramacı Vakfı

Özel Bilkent Laboratuvar Lisesi

Mathematics Portfolio HL Type I

Topic

PATTERNS FROM COMPLEX NUMBERS

Name – Surname

Selim TEPELER

IB No:

DPR160 (006040-047)

Introduction

In this portfolio we are gonna use complex number patterns which are related to roots of complex number , and we connect them our knowledge about analytic geometry ( which is distance formula) . Finally we will try to formulate a conjecture. This conjecture will help us to finding distance between roots.

PART A

Use de Moivre’s theorem to obtain solutions to the equation

Questions asks ;

 and firstly we should turn it into complex form;

 

 

De Moivre’s theorem claims that ;

  where k = 0, 1, 2 we found ;

- Cis(0) -cis(120) -cis(240) (by the way cis(α) means ‘’ cos(α)+isin(α))

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  • Use graphing software to plot these roots on an Argand diagram as well as a unit circle with centre origin.

  • Choose a root and draw line segments from this root to the other two roots.

  • Measure these line segments and comment on your results.

As we see that the length of line segments are equal to each other. (one segment is 0.01 unit bigger than the others because of accuracy of graphing software)

Repeat the above for the equations

 and

  •  Comment on your results and try to formulate a ...

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