(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, cellOutline, groupLikeTitle, center, M18, O486, R65535, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, bold, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, B65535, 12, "Calculus"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M36, N23, O486, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M36, N23, O486, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, O486, l19, o2, w378, h214, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = leftheader, inactive, M18, O486, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, O486, 12, "Times"; fontset = leftfooter, inactive, M18, O486, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M18, O486, bold, R21845, G21845, B21845, 12, "Calculus"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, R21845, G21845, B21845, 12, "Calculus"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; paletteColors = 128; currentKernel; ] :[font = input; locked; initialization; closed; preserveAspect; groupLikeNormal] *) Off[General::spell]; Off[General::spell1]; Off[Plot::plnr]; Off[ParametricPlot::ppcom]; Needs["Graphics`Colors`"] Needs["Miscellaneous`Units`"] Needs["Algebra`Trigonometry`"] Clear[Derivative] ThreeAxes[u,v] ThreeAxes[u_,v_] := Graphics3D[{ {Blue,Line[{{-u,0,0},{u,0,0}}]}, Text["x",{u + v,0,0}], {Blue,Line[{{0,-u,0},{0,u,0}}]}, Text["y",{0, u + v,0}], {Blue,Line[{{0,0,0},{0,0,u}}]}, Text["z",{0,0,u + v}]}]; ThreeAxes[u_] := ThreeAxes[u,u/8]; ThreeAxes::usage = "ThreeAxes[a,b] makes a standard cartesian axis graphics object with x, y, and z running from -a to a, and with axis labels b units beyond the tips of the axes. ThreeAxes[a] is ThreeAxes[a,a/8]."; CMView = {2.7, 1.6, 1.2}; (* :[font = input; preserveAspect; center] Miniproject For The Mathematica Workshop ;[s] 5:0,0;2,1;23,2;34,1;44,0;47,-1; 3:2,12,10,Courier,1,12,0,0,0;2,26,21,New York,1,24,0,0,0;1,26,21,New York,3,24,0,0,0; :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 143; pictureTop = 4; pictureWidth = 214; pictureHeight = 214; pictureID = 3501] :[font = subsection; inactive; preserveAspect; fontSize = 14] Gene Bild :[font = subsection; inactive; preserveAspect; fontSize = 14] The Geometry of Multiplying Complex Numbers :[font = subsection; inactive; pageBreakBelow; dontNoPageBreakBelow; preserveAspect; fontSize = 14] Brief Description: A complex number z = x + iy can be written as z = r ( cos q + i sin q ). The number r is the modulus of z, and the angle q is the argument of z. This notebook shows graphically that when two complex numbers are multiplied, their arguments are added. ;[s] 12:0,0;19,1;77,2;78,1;87,2;94,1;114,3;121,1;142,2;144,1;152,3;160,1;274,-1; 4:1,30,22,Calculus,1,14,21845,21845,21845;6,26,19,Calculus,1,12,21845,21845,21845;3,18,13,Symbol,1,12,21845,21845,21845;2,26,19,Calculus,3,12,21845,21845,21845; :[font = text; inactive; preserveAspect] Given two complex numbers z1 and z2, the first two 'do loops' display their arguments as adjacent sectors of a circle whose angles are the arguments of the numbers, with z1 in orange and z2 in light blue. The union of these two sectors forms a larger sector whose angle is the sum of the arguments of z1 and z2. The third and last 'do loop' displays the adjacent sectors representing the arguments of z1 and z2 and has a third violet sector representing the argument of z3 = z1*z2. The violet sector's angle grows from zero to the argument of z3, and exactly covers the light blue and orange sectors, showing that, in mathematical notation, Arg[z1] + Arg[z2] = Arg[z1*z2]. Note that in the second 'do loop' the angle of the light blue sector grows from the argument of z1 to the sum of the arguments of the two complex numbers z1 and z2, while in the third 'do loop' the angle of the violet sector stops at the argument of z3, so the result is driven by mathematics. If complex numbers of large modulii are input, you should adjust the PlotRange options in each of the 'do loops' accordingly. ;[s] 1:0,1;1109,-1; 2:0,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,0,12,21845,21845,21845; :[font = subsubtitle; inactive; preserveAspect] :[font = input; preserveAspect] Needs["Graphics`Colors`"]; Clear[x1,y1,z1,z2,z3,x2,y2,x3,y3,t] z1 = (*CHANGE THE NUMBER BELOW THIS COMMENT*) -.5-2I; x1 = Re[z1]; y1 = Im[z1]; z2 = (*CHANGE THE NUMBER BELOW THIS COMMENT*) 1.8+I; x2 = Re[z2]; y2 = Im[z2]; z3 = (z1)(z2); x3 = Re[z3]; y3 = Im[z3]; Do[Show[Graphics [{CadmiumOrange,Disk[{0,0},Abs[z3],{0,za}]}], AxesLabel->{"x","iy"}, Axes->True, AxesOrigin->{0,0}, Ticks->None, PlotRange->{{-5,5},{-5,5}}, AspectRatio->Automatic, DisplayFunction->$DisplayFunction], {za,0,Mod[Arg[z1],2 Pi], Mod[Arg[z1], 2 Pi]/6}]; Do[Show[Graphics [{CadmiumOrange,Disk[{0,0},Abs[z3], {0,Mod[Arg[z1],2 Pi]}], BlueLight,Disk[{0,0},Abs[z3], {Mod[Arg[z1], 2 Pi],zb}]}], AxesLabel->{"x","iy"}, Axes->True, AxesOrigin->{0,0}, Ticks->None, PlotRange->{{-5,5},{-5,5}}, AspectRatio->Automatic, DisplayFunction->$DisplayFunction], {zb,Mod[Arg[z1], 2 Pi], Mod[Arg[z2], 2 Pi] + Mod[Arg[z1],2 Pi], Mod[Arg[z2], 2 Pi]/6}] :[font = input; preserveAspect] Do[Show[Graphics [{CadmiumOrange,Disk[{0,0},Abs[z3], {0,Mod[Arg[z1], 2 Pi]}], BlueLight,Disk[{0,0},Abs[z3], {Mod[Arg[z1], 2 Pi],Mod[Arg[z1]+Arg[z2], 2 Pi]}], Violet,Disk[{0,0},Abs[z3],{0,zc}]}], AxesLabel->{"x","iy"}, Axes->True, AxesOrigin->{0,0}, Ticks->None, PlotRange->{{-5,5},{-5,5}}, AspectRatio->Automatic, DisplayFunction->$DisplayFunction], {zc,0,Mod[Arg[z3], 2 Pi], Mod[Arg[z3], 2 Pi]/8}] ^*)