1. the Further Mathematics network www.fmnetwork.org.uk

2. the Further Mathematics network www.fmnetwork.org.uk 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 re jθ are: r 1/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 z 3 =1

8. Try z 4 =1 Argand diagram?

9. nth roots of unity

10. Z n =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 re jθ are: r 1/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.