The document discusses nth roots of unity. It states that the solutions to equations of the form zn = ±1 are the nth roots of unity. When placed on an Argand diagram, they form a regular n-sided polygon with vertices on the unit circle. It provides an example of the 5th roots of unity for the equation z5 = 1. The solutions are expressed as cis(2πk/5) for k = 0, ±1, ±2. It also shows that if ω is a root of z5 - 1 = 0, then ω2, ω3, ω4, and ω5 are also roots.
2. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
3. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
4. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
5. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
6. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis
5 5 5 5
7. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis y
5 5 5 5
x
8. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis y
5 5 5 5
cis0
x
9. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis y 2
5 5 5 5 cis
5
cis0
x
10. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis y 2
5 5 5 5 cis
5
cis0
x
2
cis
5
11. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis 4 y 2
5 5 5 5 cis cis
5 5
cis0
x
2
cis
5
12. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis 4 y 2
5 5 5 5 cis cis
5 5
cis0
x
4
cis 2
5 cis
5
13. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis 4 y 2
5 5 5 5 cis cis
5 5
cis0
x
4
cis 2
5 cis
5
14. nth Roots Of Unity z n
1
The solutions of equations of the form, z n 1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5 1
2 k 0
z cis k 0, 1, 2
5
2 2 4 4
z cis 0, cis , cis , cis , cis 4 y 2
5 5 5 5 cis cis
5 5
cis0
x
4
cis 2
5 cis
5
15. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
16. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1
17. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
18. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1
19. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
20. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
21. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
12
1
22. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
12
1
2 is a solution
23. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
3 5 5 3
12
1
2 is a solution
24. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
3 5 5 3
12 13
1 1
2 is a solution
25. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
3 5 5 3
12 13
1 1
2 is a solution 3 is a solution
26. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
3 5 5 3
4 5 5 4
12 13
1 1
2 is a solution 3 is a solution
27. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
3 5 5 3
4 5 5 4
12 13 14
1 1 1
2 is a solution 3 is a solution
28. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
3 5 5 3
4 5 5 4
12 13 14
1 1 1
2 is a solution 3 is a solution 4 is a solution
29. b) i If is a complex root of z 5 1 0, show that 2 , 3 , 4 and 5 are
the other roots.
z5 1 If is a solution then 5 1
15
5 5
1 is a solution
5
2 5 5 2
3 5 5 3
4 5 5 4
12 13 14
1 1 1
2 is a solution 3 is a solution 4 is a solution
Thus if is a root then 2 , 3 , 4 and 5 are also roots