Transcript FP2 MEI Lesson 3 Complex numbers part 3_complex roots and

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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.