Math Ia

Topics: Complex number, Circle, Curve Pages: 12 (1166 words) Published: November 20, 2012
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, ( ) ( )




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...
Continue Reading

Please join StudyMode to read the full document

You May Also Find These Documents Helpful

  • Manipulatives: a Hands-on Approach to Math. Article Review Essay
  • Essay on Ia Math
  • Math Ia Essay
  • Math IA Essay
  • Essay about Math Ia
  • Math Essay
  • Essay about Math
  • math Essay

Become a StudyMode Member

Sign Up - It's Free