Math Ia
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
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