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 (| | ) ( (| | ) ) | | ( ) If we rewrite the equation with the found value of , it shows (| | ( ( ( ( ) )) )) Let k be 0, 1, and 2. When k is 0, ( ) ( ) v v Now I know that if I apply this equation with the roots of ( ) ( ) we can find the answers on the unit circle. I plotted these values in to the graphing software, GeoGebra and then I got a graph as below:
Figure 1 The roots of z-1=0 I chose a root of and I tried to find out the length of two segments from the point Z. I divided each triangle in to two same right angle triangles. By knowing that the radius of the unit circle is 1, with the knowledge of the length from D or Z to their mid-point C is length of the segment segment ) v , I found out . So I multiplied this answer by 2. And I got the v . I used same method to find out the length of the . (v v Figure 2 The graph of the equation z^3-1=0 after finding out line segment Thus we can write that the three roots of , and we can also factorize the equation by long division.
Since I know that one of the roots is 1, I can divide the whole equation by (z-1). And then I got . So if we factorize the equation as: ( )( ) As question asks I repeat the work above for the equations . Using De Moivre’s theorem, can be rewritten as: ( ) Suppose So the roots of the equation are . As we can see the graph below, I drew a graph of the roots and connected two other from a point A. The question wants me to find out the length of the line segments which I connected from a single roots to two other roots, . Since are isosceles right-angle triangles with two sides of 1.
With the basic knowledge of right triangle with two I found out that the length of the v v Figure 3 Graph of z^4-1=0 before finding out the line segment Figure 4 Graph of z^4-1=0 after finding out the line segments Again I am finding out the roots of ( ( ( Suppose that the k is equal to 0,1,2,3 and 4. ) ) ) ( ( ( ( I plotted those roots of the equation ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) in to GeoGebra and on an Argand Diagram. And as shown below I found out the length of the line segments Figure 5 Graph of z^5-1=0 before finding out the line segments Figure 6 Graph of z^5-1=0 after finding out the line segments
So if I rewrite the lengths of line segments for each different equations and , they are: , ( ) ( ) , | | | | | ( ( ( ( )| )| ( ) )| ( ) )| ( )| With my values of distance of the line segments between the chosen root and others, I made a conjecture that says ( | ( | | ( [ ]) |) ( | ( |) ) ” I tried to prove this conjecture. But as shown below, it is impossible to prove due to unknown amount of multiple of the sin properties ( ) Then I tried to prove it by binominal expansion, which is totally different way. I drew a graph of an equation (shown below) and connected between a root to all the other roots.
Figure 7 The graph of z^n-1=0, with its roots connected As shown above, the graph has certain amount of roots, and they are connected to a root as told in the problems. And the lengths of those line segments are able to be written as ( So I rewrote the equation ( And with the knowledge of ( )( )( )( ) ( ) )( ) in the form of )( )( ) ( ) And since the angles , And I will have ( ) And then, with the binominal expansion, I folded it out, and got ( ( ( ( ) ( ) ) )( )( )( ) ( ) ) ( ) And I can find out that ( ) ( ( ( )( ) )( ) )( ( ) ( ) ) And I know that ( ) , so with this knowledge, I rewrote ( ( )( ) )( ( )( ( ) ) And all those ( to zero. So it finally has )and ( ) refer ( ( ( ( ( ) ( )( )( )( ) ) ) ( ( ) ) ) ) And there are two condition where n can be even number or odd number, And according to this condition the value of n ( { ) | | | | So the total product of the length of the line segment equal to the power of the equation Proved. And I factorized When I factorized ( ( ( )( )( , I got the answers like: )( ) ) ) And I also tried to test my conjecture with some more values of For ( ) Suppose ( ) ( ) ( ( ) ) Figure 8 The graph of z^6-1=0 with line segments The product of lengths of the line segments are v v
For ( Suppose ) Figure 9 The graph of z^7-1=0 with its line segments Part B I am going to find the solutions of this equation for each Moivre’s theorem to obtain solutions to the equation . And I will use de . And I also drew diagrams for each roots of the equation s. I used Geo Gebra to represent each roots of the equation on the Argand Diagram. So, when ( ) ( ( ) ) ( ( ( ) ) ) ( ( ) ) ( ) ( ) v v ( ) ( ) Figure 10 The graph of roots of equation z^3=i As shown above, the equation has three distinct roots. 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 v 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 product
