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, ( ) ( )

√

√

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 ) √

, I found out

. So I multiplied this answer by 2. And I got the

√ . I used same method to find out the length of the . (√ √

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 √ √

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...