X2 T01 07 nth roots of unity (2011)

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<ul><li><p>nth Roots Of Unity 1nz</p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>When placed on an Argand Diagram, they form a regular n </p><p>sided polygon, with vertices on the unit circle. </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>2 0 0, 1, 25kz cis k </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>y</p><p>x</p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>y</p><p>x0cis</p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>y</p><p>x0cis</p><p>52cis</p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>y</p><p>x0cis</p><p>52cis</p><p>52cis</p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>y</p><p>x0cis</p><p>52cis</p><p>52cis</p><p>54cis</p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>y</p><p>x0cis</p><p>52cis</p><p>52cis</p><p>54cis</p><p>54cis</p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>y</p><p>x0cis</p><p>52cis</p><p>52cis</p><p>54cis</p><p>54cis</p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p>nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz</p><p>1 .. 5 zgeWhen placed on an Argand Diagram, they form a regular n</p><p> sided </p><p>polygon, with vertices on the unit circle. </p><p>y</p><p>x0cis</p><p>52cis</p><p>52cis</p><p>54cis</p><p>54cis</p><p>2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5</p><p>z cis cis cis cis cis </p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 </p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>112</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>solutionais 2112</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>solutionais 2112</p><p> 3553 </p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>solutionais 2112</p><p> 3553 113</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>solutionais 2112</p><p> 3553 </p><p>solutionais3113</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>solutionais 2112</p><p> 3553 </p><p>solutionais3113</p><p> 4554 </p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>solutionais 2112</p><p> 3553 </p><p>solutionais3113</p><p> 4554 114</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>solutionais 2112</p><p> 3553 </p><p>solutionais3113</p><p> 4554 </p><p>solutionais4114</p></li><li><p> roots.other he t</p><p> are and ,, that show ,01 ofroot complex a is If b) 54325 zi</p><p>15 z 1hen solution tais If 5 11555</p><p>solutionais5 2552 </p><p>solutionais 2112</p><p> 3553 </p><p>solutionais3113</p><p> 4554 </p><p>solutionais4114</p><p>roots also are and ,,root then a is if Thus 5432 </p></li><li><p> 0 1 that Prove 432 ii</p></li><li><p> 0 1 that Prove 432 iiab 5432 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 1</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 OR</p><p>2 3 41 1</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 OR</p><p>2 3 41 GP: 1, , 5a r n 1</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 OR</p><p>2 3 41 GP: 1, , 5a r n 11</p><p>na rr</p><p> 1</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 OR</p><p>2 3 41 GP: 1, , 5a r n 11</p><p>na rr</p><p> 51 11</p><p> 1</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 OR</p><p>2 3 41 GP: 1, , 5a r n 11</p><p>na rr</p><p> 51 11</p><p> 0</p><p> 1</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 OR</p><p>2 3 41 GP: 1, , 5a r n 11</p><p>na rr</p><p> 51 11</p><p> 0 5 1 0 </p><p> 1</p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 1</p><p> 324 and are roots hoseequation w quadratic theFind iii</p><p>OR2 3 41 GP: 1, , 5a r n 1</p><p>1</p><p>na rr</p><p> 51 11</p><p> 0 5 1 0 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 1</p><p> 324 and are roots hoseequation w quadratic theFind iii324 </p><p>OR2 3 41 GP: 1, , 5a r n 1</p><p>1</p><p>na rr</p><p> 51 11</p><p> 0 5 1 0 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 1</p><p> 324 and are roots hoseequation w quadratic theFind iii324 </p><p>1</p><p>OR2 3 41 GP: 1, , 5a r n 1</p><p>1</p><p>na rr</p><p> 51 11</p><p> 0 5 1 0 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 1</p><p> 324 and are roots hoseequation w quadratic theFind iii324 </p><p>1 324 </p><p>OR2 3 41 GP: 1, , 5a r n 1</p><p>1</p><p>na rr</p><p> 51 11</p><p> 0 5 1 0 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 1</p><p> 324 and are roots hoseequation w quadratic theFind iii324 </p><p>1 324 </p><p>7643 </p><p>OR2 3 41 GP: 1, , 5a r n 1</p><p>1</p><p>na rr</p><p> 51 11</p><p> 0 5 1 0 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 1</p><p> 324 and are roots hoseequation w quadratic theFind iii324 </p><p>1 324 </p><p>7643 </p><p>1</p><p>243</p><p>OR2 3 41 GP: 1, , 5a r n 1</p><p>1</p><p>na rr</p><p> 51 11</p><p> 0 5 1 0 </p></li><li><p> 0 1 that Prove 432 iiab 5432 rootstheof sum</p><p>01 432 15 OR 015 </p><p> 011 432 </p><p> 01101</p><p>:</p><p>12 n</p><p>n</p><p>NOTE</p><p>01 432 1</p><p> 324 and are roots hoseequation w quadratic theFind iii324 </p><p>1 324 </p><p>7643 </p><p>1</p><p>243</p><p>01 isequation 2 xx</p><p>OR2 3 41 GP: 1, , 5a r n 1</p><p>1</p><p>na rr</p><p> 51 11</p><p> 0 5 1 0 </p></li><li><p>c) If is a complex cube root of unity, use the fact that to; 01 2 </p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 3</p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii </p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p><p>21</p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p><p>21</p><p>1</p><p>2</p><p>2</p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p><p>21</p><p>1</p><p>2</p><p>2</p><p>(iii) Form the cubic equation with roots 21 ,1 ,1 </p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p><p>21</p><p>1</p><p>2</p><p>2</p><p>(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz</p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p><p>21</p><p>1</p><p>2</p><p>2</p><p>(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz</p><p> 011 2 zzz</p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p><p>21</p><p>1</p><p>2</p><p>2</p><p>(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz</p><p> 011 2 zzz01223 zzzzz</p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p><p>21</p><p>1</p><p>2</p><p>2</p><p>(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz</p><p> 011 2 zzz01223 zzzzz0122 23 zzz</p></li><li><p> 32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 </p><p> 31</p><p>3</p><p>2</p><p>1 1( ) Evaluate 1 1</p><p>ii 11</p><p>2 </p><p>21</p><p>1</p><p>2</p><p>2</p><p>(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz</p><p> 011 2 zzz01223 zzzzz0122 23 zzz</p><p>Exercise 4I; 1, 3, 5 (need 4b), 7</p><p>Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Slide Number 50Slide Number 51Slide Number 52Slide Number 53Slide Number 54Slide Number 55Slide Number 56Slide Number 57Slide Number 58Slide Number 59Slide Number 60Slide Number 61Slide Number 62Slide Number 63Slide Number 64Slide Number 65</p></li></ul>