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

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

3. 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.

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

5. Solve z3=1

8. Try z4=1 Argand diagram?

9. nth roots of unity

10. Zn =1

11. Sum of cube roots?

16. Find the four roots of -4

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

18. Example:

20. 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.

21. 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.