fp2 (mei) complex numbers- complex roots and geometrical interpretations

the Further Mathematics network

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the Further Mathematics network

www.fmnetwork.org.uk

FP2 (MEI)Complex Numbers-

Complex roots and geometrical interpretations

Let Maths take you Further…

Complex roots and geometrical interpretationsBefore 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 =1

)sin(cos i

Sum of cube roots?

rnr )*(

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 interpretationsBefore 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.