x2 t01 07 nth roots of unity (2012)

nth Roots Of Unity 1nz

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

2 0 0, 1, 25kz cis k

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

y

x

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

y

x0cis

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

y

x0cis

52cis

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

y

x0cis

52cis

52

cis

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

y

x0cis

52cis

52

cis

54cis

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

y

x0cis

52cis

52

cis

54cis

54

cis

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

y

x0cis

52cis

52

cis

54cis

54

cis

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz

1 .. 5 zge

When placed on an Argand Diagram, they form a regular n

sided polygon, with vertices on the unit circle.

y

x0cis

52cis

52

cis

54cis

54

cis

2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5

z cis cis cis cis cis

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

11555

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

11555

solutionais5

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

11555

solutionais5

2552

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

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solutionais5

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112

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

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solutionais5

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solutionais 2112

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

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solutionais5

2552

solutionais 2112

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roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

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solutionais5

2552

solutionais 2112

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113

roots.other he t

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15 z 1hen solution tais If 5

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solutionais 2112

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roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

11555

solutionais5

2552

solutionais 2112

3553

solutionais3113

4554

roots.other he t

are and ,, that show ,01 ofroot complex a is If b) 54325 zi

15 z 1hen solution tais If 5

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solutionais5

2552

solutionais 2112

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114

roots.other he t

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roots.other he t

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15 z 1hen solution tais If 5

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roots also are and ,,root then a is if Thus 5432

0 1 that Prove 432 ii

0 1 that Prove 432 ii

ab

5432

0 1 that Prove 432 ii

ab

5432 rootstheof sum

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432 1

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432

OR2 3 41

1

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432

OR2 3 41 GP: 1, , 5a r n

1

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

1

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

1

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0

1

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0 5 1 0

1

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432 1

324 and are roots hoseequation w quadratic theFind iii

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0 5 1 0

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432 1

324 and are roots hoseequation w quadratic theFind iii324

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0 5 1 0

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432 1

324 and are roots hoseequation w quadratic theFind iii324

1

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0 5 1 0

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432 1

324 and are roots hoseequation w quadratic theFind iii324

1 324

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0 5 1 0

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432 1

324 and are roots hoseequation w quadratic theFind iii324

1 324

7643

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0 5 1 0

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432 1

324 and are roots hoseequation w quadratic theFind iii324

1 324

7643

1

243

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0 5 1 0

0 1 that Prove 432 ii

ab

5432 rootstheof sum

01 432 15

OR 015 011 432

011

01:

12 n

n

NOTE

01 432 1

324 and are roots hoseequation w quadratic theFind iii324

1 324

7643

1

243

01 isequation 2 xx

OR2 3 41 GP: 1, , 5a r n 1

1

na rr

51 1

1

0 5 1 0

c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

21

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

21

1

2

2

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

21

1

2

2

(iii) Form the cubic equation with roots 21 ,1 ,1

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

21

1

2

2

(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

21

1

2

2

(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz

011 2 zzz

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

21

1

2

2

(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz

011 2 zzz01223 zzzzz

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

21

1

2

2

(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz

011 2 zzz01223 zzzzz0122 23 zzz

32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω

3

1

3

2

1 1( ) Evaluate 1 1

ii

11

2

21

1

2

2

(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz

011 2 zzz01223 zzzzz0122 23 zzz

Exercise 4I; 1, 3, 5 (need 4b), 7