the Further Mathematics network
www.fmnetwork.org.uk
the Further Mathematics network
www.fmnetwork.org.uk
FP2 (MEI)
Complex NumbersComplex 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.
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.
Recap: Euler’s relation and De Moivre
De Moivre:

Solve z3=1

Try z4=1
Argand diagram?
nth roots of unity

Zn
(cos   i sin  )
=1

Sum of cube roots?
( )*  
r
nr

Find the four roots of -4
Geometrical uses of complex numbers
Loci from FP1 (in terms of the argument of a complex number)
Example:
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.
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.