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Math Portfolio Type 2 - PATTERNS FROM COMPLEX NUMBERS

Extracts from this document...

Introduction

image43.jpg

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 image00.png

Questions asks ; image01.png

 and firstly we should turn it into complex form;

image02.png

De Moivre’s theorem claims that ;

image44.jpg  where k = 0, 1, 2 we found ;

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

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

image49.png

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

image50.png

  • Measure these line segments and comment on your results.

image51.png

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 image03.png

 and image04.png

  •  Comment on your results and try to formulate a conjecture.
...read more.

Middle

 value cos(0) ,  image11.png

 is cos(72) image12.png

 value is sin(0) image13.png

 value is sin(72).

Now we will put them on equation .

image14.png

image15.png

1+1-2(cos72*cos 0 +sin72*sin0)

We will use this trigonometric identities ;

image54.png and

image55.png

When we put our value at these formula ; image16.png

=cos 72

Lastly our final formula is ;

image17.png

Factorize image18.png

  •  for n = 3, 4 and 5.

image19.png

-1= (z-1)(image20.png

image21.png

-1= (z-1)(image22.png

image23.png

-1= (z-1)(image24.png

  • Use graphing software to test your conjecture for some more values of n € Z+ and make modifications to your conjecture if necessary.

Lets try it for n=6

image56.png

Testing for Aw distance ;

 a=cis0 and w is cis120

image17.png

image25.png

=1.732050… ≅17.3 (because of accuracy of software)

or testing for Az distance ;

image17.png

image26.png

=1= 1 from graph .

  • Prove your conjecture.

PART B

...read more.

Conclusion

=i

image44.jpg  where where k = 0, 1, 2,3,4

We found ;

Cis(18) cis (90) cis(162) cis(234) cis(306)

  • Use graphing software to represent each of these solutions on an Argand diagram.

For image28.png

=i

image45.png

For image29.png

=i

image46.png

For image30.png

=i

image47.png

Generalize and prove your results for i  image27.png

  • =a + bi , where |a +bi |=1.

|a +bi |=1 so it means that ;

image31.png

+image32.png

1=image33.png

As we can see r=1 so  image27.png

=a+bi= r cis(θ)

When we use De Moivre’s theorem, it states ;

image48.jpg

For our solution we will use reverse way of it ;

image34.png

 r=1 so we dont need r at equation ;

z= image35.png

let’s find the image36.png

 value with tangent ;

tan(image37.png

=image38.png

image36.png

 = image39.png

  and  image40.png

=image41.png

   so z=cis(image42.png

)

  • What happens when |a +bi |≠1?

That doesn’t apply when when |a +bi |≠1 because r value should equal to 1.

Resources

http://en.wikipedia.org/wiki/Distance

http://en.wikipedia.org/wiki/List_of_trigonometric_identities

http://www.rapidlearningcenter.com/mathematics/trigonometry/18-Complex-Numbers-and-De-Moivre-Theorem.html

http://demonstrations.wolfram.com/PatternsFromMathRulesUsingComplexNumbers/

http://en.wikipedia.org/wiki/Imaginary_number

...read more.

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