Fundamental Mathematics from an Advanced Standpoint

Math 4303

Helmut Knaust, Department of Mathematical Sciences, UTEP, El Paso TX 79968
hknaust@utep.edu

Last Edits: 9/29/2009

Multiplication of Complex Numbers

The red arrow depicts the product of the two complex numbers shown in green.

w=RandomReal[{1,2}]Exp[I RandomReal[{0,2π}]];Manipulate[Show[Graphics[{{Dotted,Circle[{0,0},Abs[a(Cos[b]+ I Sin[b])w]]},{AbsoluteThickness[3],Green,Arrow[{{0,0},{Re[w],Im[w]}}]},{AbsoluteThickness[3],Green,Arrow[{{0,0},{a Cos[b],a Sin[b]}}]},{AbsoluteThickness[3],Red,Arrow[{{0,0},{Re[a(Cos[b]+ I Sin[b])w],Im[a(Cos[b]+I Sin[b])w]}}]}}],Axes->True,BaseStyle->{14,Bold},PlotRange->{{-5,5},{-5,5}}],{{a,1,"r"},.5,2},{{b,0,"θ"},0,2π}]

The (n^th) roots of a complex number

NthRoots[z_,n_]:=Module[{sol,w,h,t},h=Mod[n,100];t=Mod[n,10];If[n>50,Print["The number n is quite large. Solutions may not be accurate."]];sol=NSolve[w^n==z,w];Show[Graphics[{{Dotted,Circle[{0,0},Abs[z]^(1/n)]},{AbsolutePointSize[7],Blue,Map[(Point[#]&),Transpose[{Re[w]/.sol,Im[w]/.sol}]]}}],Axes->True,BaseStyle->{14,Bold},PlotLabel->Row[{"The ",n,Which[t==1&&Not[h==11],Superscript["","st"],t==2&&Not[h==12],Superscript["","nd"],t==3&&Not[h==13],Superscript["","rd"],True,Superscript["","th"]]," roots of ",z}]]]

NthRoots[100-200I,3]

Manipulate[NthRoots[2-5I,k],{k,1,20,1}]