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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 49 "High School Modul
es > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 "    " 
}{TEXT 256 26 "Imaginary &Complex Numbers" }}{PARA 0 "" 0 "" {TEXT -1 
196 "\nAn exploration of the derivation of imaginary and complex numbe
rs, numerical operations with imaginary and complex numbers, complex c
onjugates, and graphing complex numbers on the complex plane.\n" }}
{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the Code Resource
 section first. Although there will be no output immediately, these de
finitions are used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 " 0. Code" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 673 "ComplexPlot  :=  proc( z )\nlocal \+
a,b,c,d,Axes,Pt, Run, Rise;\nif (Re(z) = 0) then  b := 1; else b := 1.
3*abs(Re(z));   fi;  \na :=  - b;\nif (Re(z) > 0) then a := a/2; else \+
b := b/2; fi;\n\nif (Im(z) = 0) then  d := 1; else d := 1.3*abs(Im(z))
;  fi;   \nc :=  - d;\nif (Im(z) > 0) then  c:= c/2; else d :=d/2;  fi
;   \n\nAxes := plot(0, x = a..b, y = c..d, color = black, scaling = c
onstrained):\n\nPt := plottools[disk]([ Re(z), Im(z) ], 1/3, color=red
 ):\n    \nRun := plot( [[ 0, Im(z)], [Re(z), Im(z)]], color = blue, l
inestyle = 2 );\nRise := plot( [[ Re(z), 0],[Re(z), Im(z)] ], \n      \+
                  color = blue, linestyle = 2 );\n\nplots[display](Axe
s,Pt, Run, Rise);\nend proc:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 
" 1.  The Origin of Imaginery Numbers" }}{PARA 0 "" 0 "" {TEXT -1 77 "
\nThere are no real numbers, which upon being squared yield negative n
umbers.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x^2 = - 1;  " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x^2 = - 9; " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x^2 = - 12; " }}}{PARA 0 "" 0 "" 
{TEXT -1 99 "\nTo solve these we can \"invent\" a new number that will
 help us make some sense of these equations.\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "sqrt(-1);" }}}{PARA 0 "" 0 "" {TEXT -1 241 "\nThi
s simple definition gives us a way to solve the kinds of problems we s
aw above. When we solve equations with positive constants like this, w
e get both a plus and minus answer. And so it is with solving equation
s with negative constants.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "x^2 = + 25; solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "x^2 = - 1; solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"x^2 = - 9; solve(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x^
2 = - 12; solve(%);" }}}{PARA 0 "" 0 "" {TEXT -1 74 "\nNote that i (I \+
in Maple) has the property that when squared it gives -1. " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "I^2;" }}}{PARA 0 "" 0 "" {TEXT -1 
118 "\nClearly this is not a real number since no real numbers which h
ave this property. We call numbers are multiples of i " }{TEXT 259 18 
"imaginary numbers." }{TEXT -1 368 " This name is not to be taken too \+
literally. Gauss, one of the originators of imaginary numbers, later w
ished that they be called something else to avoid the whimsical imager
y of that name. However, the name has taken hold and we are stuck with
 it now. Although it is not evident now, rest assured that imaginary n
umbers are used in many, many real world applications." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`Examples of imaginary uumbers :`;I
; 3*I; -13*I; sqrt(17)*I;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 32 " 2.  Powers of Imaginery Numbers" }}
{PARA 0 "" 0 "" {TEXT -1 211 "\nIf we look at the powers of imaginary \+
numbers, we will see some interesting patterns. The first four have fo
ur distinct values. However, if we look at the higher powers we see th
at the same four values repeat.\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "I^k $ k = 1..4 ;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "I^k $ k = 1..20 ;" }}}{PARA 0 "" 0 "" {TEXT -1 78 "\n
In fact, i, to any power ( of a natural number) is one of these four v
alues.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "I^(111); I^(222)
; I^(333); I^(444);" }}}{PARA 0 "" 0 "" {TEXT -1 200 "\n\nWhat about n
egative exponents? Since 1/i = (i^3)/(i^4) = i^3/1 = i^3 = -i, the neg
ative exponents simply give the negative of the value given by the pos
itive exponent. Thats really unusual isn't it?" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 4 "1/I;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "I^(-k) $ k = 1..20 ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 34 " 3.  The Origin of Complex Numbers" }}
{PARA 0 "" 0 "" {TEXT -1 138 "\nRecall that when we have a quadratic e
quation, and use the quadratic formula to solve it, we end up with a s
quare root of D = b^2 - 4ac.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "a*x^2 + b*x + c = 0; solve( %, x);" }}}{PARA 0 "" 0 "" {TEXT -1 
46 "\nThis works out well if D is positive or zero." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 62 "3*x^2 + 7*x - 2 = 0;  \nD = 7^2 - 4*3*(-2);
 \nx = solve( %%, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "5*
x^2 - 30*x + 45 = 0; \nD = (-30)^2 - 4*5*45; \nx = solve( %%, x);" }}}
{PARA 0 "" 0 "" {TEXT -1 118 "\n\n However, if D is negative, then thi
s means that we'll have the square of a negative number as part of the
 solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "x^2 + 3*x + 5 \+
= 0;  \nD = 3^2 - 4*1*5; \nx = solve( %%, x);" }}}{PARA 0 "" 0 "" 
{TEXT -1 59 "\nNumbers which are part real and part imaginary are coll
ed " }{TEXT 260 15 "complex numbers" }{TEXT -1 110 ". They are neither
 purely real, nor purely imaginary - they are a combination of both. H
ere are some examples." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "3 +
 4*I;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "-13 + I;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "41 - 10*sqrt(2)*I;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "1 + Pi*I;" }}}{PARA 0 "" 0 "
" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 " 4.  Operat
ions with Complex Numbers" }}{PARA 0 "" 0 "" {TEXT -1 129 "\nWe can tr
eat complex numbers just as we treat real numbers and add, subtract, m
ultiply or divide them - with a few precautions.\n" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT 263 37 "       Complex Addi
tion & Subtraction" }}{PARA 0 "" 0 "" {TEXT -1 226 "\nNotice that the \+
real parts combine and imaginary parts combine, but two don't mix. Its
 much the same as combining like terms. Note that its possible to add \+
two complex numbers and get a purely real or purely imaginary number.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(10 + 20*I) + (3 + 7*I);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(4 - 4*I) + (-11 + 6*I)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "(10 + 20*I) + (-10 - 8
*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(-9 + 17*I) + (9+ 1
7*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "(30 + 40*I) - (35 \+
+ 35*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "(-10 + 31*I) - \+
(10 - 28*I);" }}}{PARA 0 "" 0 "" {TEXT -1 5 "     " }}{PARA 0 "" 0 "" 
{TEXT 264 39 "        Complex Multiplication & Powers" }}{PARA 0 "" 0 
"" {TEXT -1 237 "\nComplex numbers are much like binomials. When multi
plying two binomials we would use the FOIL method, andn so it is with \+
complex numbers too. The only thing, is that I^2 = -1. And so there is
 a little more simplification at the the end." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "(4 + 5*x)*(6 + 7*x): % = expand(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "(4 + 5*I)*(6 + 7*I);" }}}{PARA 0 "
" 0 "" {TEXT -1 55 "\nNotice that 24 - 35*I^2 = 24 - 35*(-1) = 4 + 35 \+
= 58.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(13 - 4*I)*(12  +
 11*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(7  + 2*I)*(8  +
 3*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(1 + 4*I)*(1  - 2
1*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "(100 + 7*I)^2;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "(100 - 7*I)^2;" }}}{PARA 0 
"" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 " 5.  \+
Simplifying Complex Numbers " }}{PARA 0 "" 0 "" {TEXT -1 9 "\n        \+
" }{TEXT 261 19 "Conjugate Principle" }{TEXT -1 315 "\n\nNotice that i
f we multiply a complex number times its conjugate - the same number w
ith the sign on the imaginary part reversed - we get a purely real num
ber. It works much the same way that conjugates of radical expressions
 works to eliminate radical expressions - in this case I = sqrt(-1) - \+
from an expression.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "(3 \+
+ 4*I)*(3 - 4*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(17 + \+
29*I)*(17 - 29*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "(sqrt
(5) + 2*I)*(sqrt(5) - 2*I): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "(1 + 10*I)*(1 - 10*I);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "(20 + 3*I)*(20 - 3*I);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "(sqrt(3) + sqrt(5)*I)*(sqrt(3) - sqrt(5)*I): % = ex
pand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 56 "\n
\nNote that Maple can find the conjugate automatically.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "z:= 3 + 20*I;  zcon := conjugate(z)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "z*zcon;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "z:= 50 -  13*I;  zcon := conjugate(
z);z*zcon;" }}}{PARA 0 "" 0 "" {TEXT -1 5 "     " }}{PARA 0 "" 0 "" 
{TEXT 262 21 "      Complex Divsion" }}{PARA 0 "" 0 "" {TEXT -1 242 "
\nThe conjugate principle can be used to \"divide\" complex numbers. A
ctually, we'll perform an operation more akin to rationalizing the den
ominator than dividing. But the result will be the quotient nonetheles
s.\n\nMaple does this automatically.\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "(3 + 10*I)/(7 - 4*I);" }}}{PARA 0 "" 0 "" {TEXT -1 
121 " \nWe can also do it manually - by multiplying top and bottom of \+
the fraction by the complex conjugate of the denominator." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "top := 3 + 10*I;\nbot := 7 - 4*I;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "botcon := conjugate( bot \+
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "top*botcon; bot*botco
n; %%/%;" }}}{PARA 0 "" 0 "" {TEXT -1 27 " \n\nHere is another example
 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(2 + 1*I)/(7 - 4*I);  \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "top := 2 + 1*I;\nbot :=
 7 - 4*I; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "botcon := con
jugate( bot );\ntop*botcon: \nbot*botcon: \n`top `= %%, `bottom `= %; \+
\n%%%/%%;" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 48 " 6.  Geometric Interpretation of Complex Numbers" }}
{PARA 0 "" 0 "" {TEXT -1 192 "\nThere is a graphic way of representing
 complex numbers. The real part is marked off on the x axis, and the i
maginary part is marked off on the y axis. This defined a unique point
 on a plane.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "z := 3 + 4
*I;\nComplexPlot( %);  \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "z := -8 + 3*I;\nComplexPlot( %);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "z := 9 - 4*I;\nComplexPlot( %);" }}}{PARA 0 "" 0 "" 
{TEXT -1 64 "\nNotice that conjugates appear as reflections across the
 x axis." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "3 + 4*I, 3 - 4*I
;\nplots[display](ComplexPlot( 3 + 4*I),ComplexPlot( 3 - 4*I));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "-8 + 2*I, -8 - 2*I;\nplots[d
isplay](ComplexPlot( -8+ 2*I),ComplexPlot(-8 - 2*I));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "2*I, -2*I;\nplots[display](ComplexP
lot(2*I),ComplexPlot(-2*I));" }}}{PARA 0 "" 0 "" {TEXT -1 86 "\nAnothe
r interesting thing we can do is to plot two complex numbers and their
 product." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "z1 := 1 + 4*I;
 z2 := 4 - 1*I;\nplots[display](  ComplexPlot( z1),\n                 \+
ComplexPlot( z2),\n                 ComplexPlot( z1*z2) );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "z1 := 2*(1 + I)/evalf(sqrt(2)); z2
 := 3*(1 + evalf(sqrt(3))*I)/2;\nplots[display](  ComplexPlot( z1),\n \+
                ComplexPlot( z2),\n                 ComplexPlot( z1*z2
) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "z1 := 2 + 2*I; z2 \+
:= 1 - 1*I;\nplots[display](  ComplexPlot( z1),\n                 Comp
lexPlot( z2),\n                 ComplexPlot( z1*z2) );" }}}{PARA 0 "" 
0 "" {TEXT -1 2 "\n " }}}{PARA 0 "" 0 "" {TEXT 265 36 "\n         \251
 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "5
 2 0 0" 185 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 
1 1 }