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FP2 (MEI) Complex Numbers- Complex roots and geometrical interpretations

FP2 (MEI) Complex Numbers- Complex roots and geometrical interpretations. Let Maths take you Further…. Complex roots and geometrical interpretations. Before you start: • You need to have covered the chapter on complex numbers in Further Pure 1, and the work in sections 1 – 3 of this chapter.

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FP2 (MEI) Complex Numbers- Complex roots and geometrical interpretations

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  1. FP2 (MEI)Complex Numbers-Complex roots and geometrical interpretations Let Maths take you Further…

  2. Complex roots and geometrical interpretations Before you start: • You need to have covered the chapter on complex numbers in Further Pure 1, and the work in sections 1 – 3 of this chapter. When you have finished…You should: • Know that every non-zero complex number has n nth roots, and that in the Argand diagram these are the vertices of a regular n-gon. • Know that the distinct nth roots of rejθ are: r1/n [ cos((θ + 2kπ)/ n) +jsin((θ + 2kπ)/ n) ] for k = 0, 1,…, n - 1  • Be able to explain why the sum of all the nth roots is zero. • Be able to apply complex numbers to geometrical problems.

  3. Recap: Euler’s relation and De Moivre De Moivre:

  4. Solve z3=1

  5. Try z4=1 Argand diagram?

  6. nth roots of unity

  7. Zn =1

  8. Sum of cube roots?

  9. Find the four roots of -4

  10. Geometrical uses of complex numbers Loci from FP1 (in terms of the argument of a complex number)

  11. Example:

  12. Complex roots and geometrical interpretations Before you start: • You need to have covered the chapter on complex numbers in Further Pure 1, and the work in sections 1 – 3 of this chapter. When you have finished…You should: • Know that every non-zero complex number has n nth roots, and that in the Argand diagram these are the vertices of a regular n-gon. • Know that the distinct nth roots of rejθ are: r1/n [ cos((θ + 2kπ)/ n) +jsin((θ + 2kπ)/ n) ] for k = 0, 1,…, n - 1  • Be able to explain why the sum of all the nth roots is zero. • Be able to apply complex numbers to geometrical problems.

  13. Independent study: • Using the MEI online resources complete the study plan for Complex Numbers 4: Complex roots and geometrical applications • Do the online multiple choice test for this and submit your answers online.

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