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