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FP2 (MEI) Complex Numbers- Complex 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. - PowerPoint PPT Presentation
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the Further Mathematics network
www.fmnetwork.org.uk
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.