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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 50 "High School Modul
es > Geometry by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 "    \+
" }{TEXT 256 11 "The Number " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }}{PARA 0 
"" 0 "" {TEXT -1 57 "\nAn exploration of the famous number of legend a
nd lore.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the
 Code Resource section first. Although there will be no output immedia
tely, these definitions are used later in this worksheet.]" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 
260 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; w
ith(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "scan   := '
scaling = constrained, axes = none':" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 263 "r10    := proc()  2*(evalf(rand()/10^11)-5);      en
d proc: \nrr     := proc()  2+ 4*(evalf(rand()/10^12));     end proc: \+
\nshade  := proc()  local s; \n                  s := evalf(rand()/10^
12,2)/4 + .3; \n                  COLOR(RGB, s,s,.9); \n          end \+
proc: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "CP := (h,k,r) -> \+
plottools[circle]([h,k], r, thickness = 2, color = shade()):" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 14 "1. The Number " 
}{XPPEDIT 18 0 "Pi" "6#%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 146 "\nLike s
quares, but unlike triangles, all circles are similar to each other. E
ach circle is just a smaller or larger version of all other circles.\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display( seq(   CP( r10(
), r10(), rr() ), k = 1..10), scan);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "display( seq(   CP( r10(), r10(), rr() ), k = 1..10),
 scan);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display( seq(   \+
CP( r10(), r10(), rr() ), k = 1..10), scan);" }}}{PARA 0 "" 0 "" 
{TEXT -1 145 "\n\nThe ratio of the circumference (the distance around)
 to the diameter (the distance across) is the same for all of these ci
rcles. That ratio is " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 328 ". \+
Suppose we measured various round objects - both around and across, th
en took their ratios.... this is a good class project ... trashcans, s
oda cans, round lamps, amphitheatres....anything round can be measured
 roughly. The units don't matter as long as they are the same for both
 measurements.\n\nHere are some sample numbers.\n" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 40 "C := 8; d := 5/2; C/d : % = evalf(%, 5);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "C := 17; d := 16/3; C/d : % \+
= evalf(%, 5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "C := 69; \+
 d := 22; C/d : % = evalf(%, 5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "C := 300; d := 95; C/d : % = evalf(%, 5);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "C := 472; d := 150; C/d : % = evalf
(%, 5);" }}}{PARA 0 "" 0 "" {TEXT -1 3 "\n\n " }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 1 " " }{TEXT 263 24 "2. Decimal Expansion of " }
{XPPEDIT 18 0 "Pi" "6#%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 95 "\nLegend h
as it that one mathematician dedicated decades of his life to computin
g the digits of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 157 ". It's \+
an interesting tale. As recently as the 1960's, during the slide rule \+
era, a student, teacher, or engineer would often use 3.14 as an approx
imatio to " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 224 ". Although co
mputers existed and many more digits were known at the time, they were
 not readily computable by \"average\" people. In the mid-1970's, calc
ulators became available for several hundred dollars that could comput
e  " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 108 " to ten decimal plac
es or so. Even today, most graphing calculators are limited to 15 or 1
6 decimal places.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Pi = \+
evalf(Pi, 16);" }}}{PARA 0 "" 0 "" {TEXT -1 33 "\nBy the end of the 16
th century, " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 171 " was known \+
to 30 decimal places. By the end of the 18th century, it was known to \+
140 places. And by the end of the 19th century, it was known to about \+
500 accurate places." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Pi =
 evalf(Pi, 30);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Pi = eva
lf(Pi, 140);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Pi = evalf(
Pi, 500);" }}}{PARA 0 "" 0 "" {TEXT -1 118 "\n\nEven with the advent o
f personal computers in the early 1980's it was so automatic that it w
ould be easy to compute " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 142 
" to many more decimal places - because computers store data in bytes.
 A 32 byte fraction has roughly this resolution, about 10 decimal plac
es." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "1/2^32; evalf(%);" }}
}{PARA 0 "" 0 "" {TEXT -1 442 "\n\nToday, with Maple, you can see what
 was unthinkable even a few decades ago, much less several centuries a
go. Maple includes some remarkable programming that transcends the bou
nds of computer storage bytes to compute an almost unlimited number of
 decimal places. Simply choose how many decimal places of \271 you wan
t to see, and as if there were a genie granting your wishes, it will c
ome true. 100 decimal places, 1,000 places, 2,500 places. " }}{PARA 0 
"" 0 "" {TEXT -1 1 "\n" }{TEXT 270 50 "Optional Fun Section (Uses some
 Precalculus Math)\n" }{TEXT -1 44 "\nIf we consider the known decimal
 places of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 200 " as a measur
ement of technological advancement, it looks like it might have grown \+
exponentially. Lets take two \"points\" : (year 1600, 30 places) and (
year 1800, 140 places) to extrapolate a function.\n" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 135 "C := 'C': t := 't':  k := 'k':\nf := t-> C
*exp(k*t);\nEQ1 :=      f(1600) = 30;\nEQ2 :=      f(1800) = 140;\nsol
ve(EQ1, C) = solve(EQ2, C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "k := solve(%,k);\nEQ1;\nC := solve(%,C);" }}}{PARA 0 "" 0 "" 
{TEXT -1 72 "\nHere is the function which should give the number of de
cimal places of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 21 " based o
n the year.  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f(t);\nsimp
lify(f(t),exp);" }}}{PARA 0 "" 0 "" {TEXT -1 107 "\nAccording to this \+
formula, lets see how many decimal places there should be in the years
 2000 and year 0. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "floor(
evalf( f(2000) ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "floor
(evalf( f(0) ));" }}}{PARA 0 "" 0 "" {TEXT -1 98 "It seems the ancient
s were ahead of schedule! Or more likely the dark ages put us behind s
chedule." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "floor(evalf( f(1
200) ));" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 1 " " }{TEXT 261 19 "3. The History of  " }{XPPEDIT 18 0 "
Pi" "6#%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 12 "\nThe number " }{XPPEDIT 
18 0 "Pi" "6#%#PiG" }{TEXT -1 219 ", or some variation of it, has exis
ted for at least several milenia. Who knows if the inventor of the whe
el thought about the ratio of how far the wheel travels in one turn co
mpared to the width of the wheel - which is " }{XPPEDIT 18 0 "Pi" "6#%
#PiG" }{TEXT -1 29 ". However, the true value of " }{XPPEDIT 18 0 "Pi
" "6#%#PiG" }{TEXT -1 40 " did not come to be known immediately.\n\n" 
}{TEXT 264 4 "    " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 271 5 " = 3?
" }{TEXT 266 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 11 "The number " }
{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 133 " holds a special significa
nce in mathematics and in history. Various ancient peoples such as the
 Babylonians used the value of 3 for " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }
{TEXT -1 180 ". A less ancient people who believed this was one partic
ular state legislature of a state (which shall remain anonymous) about
 100 years ago, which tried to legislate the value of " }{XPPEDIT 18 
0 "Pi" "6#%#PiG" }{TEXT -1 39 ".\n\nHow accurate is this approximation
?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "3/Pi; % = evalf(%,10)
;" }}}{PARA 0 "" 0 "" {TEXT -1 154 "\nIt's about 95% accurate. If an a
ncient architect of a circular colliseum believed the radius to be 300
 feet, how many more feet of fencing would he need?" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 47 "2*300*3;\n2*300*Pi;\ndifference = evalf( % \+
- %%);" }}}{PARA 0 "" 0 "" {TEXT -1 50 "\nSimply because of the innacc
uracy of his or her \"" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 91 "\"
, the achetect would need an additional 85 feet of fencing. It must ha
ve made him wonder.\n" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 265 4 
"    " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 272 9 " = 3 1/7?" }{TEXT 
-1 23 "\n\nA \"new and improved\" " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }
{TEXT -1 85 ", was found by the ancient Egyptians. They used the value
 of 3 1/7, which is 22/7.\n  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "EPi := 22/7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "EPi/P
i;   % = evalf(%,10);" }}}{PARA 0 "" 0 "" {TEXT -1 160 "\n\nThis appro
ximation is remarkably better. It is only off by 4/100 of 1 percent! I
f the architect used this value on his coliseum, how much would he be \+
off by?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "2*300*EPi;\n2*3
00*Pi;\ndifference = evalf( % - %%);" }}}{PARA 0 "" 0 "" {TEXT -1 263 
"\nAbout 3/4 of a foot too much. That's not bad at all. What if the wa
nted to compute the distance the earth travels, using the \"facts\" th
at the sun travels in a circular orbit about the sun (actually its ell
iptical) and the radius is 10,000 miles (grossly wrong).\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "2*10000*EPi;\n2*10000*Pi;\ndifferen
ce = evalf( % - %%);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "\n \n\n" }{TEXT 
268 3 "   " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 273 12 " = 3 17/120?
" }{TEXT -1 62 "\n\nA Romans used a better value, 3 17/120, which is 3
77/120.\n  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RPi := 3 + 17
/120;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "RPi/Pi;   % = eval
f(%,10);" }}}{PARA 0 "" 0 "" {TEXT -1 161 "\n\nThis approximation is b
etter still. It is only off by about 1/425 of 1 percent! If the archet
ect used this value on his coliseum, how much would he be off by?\n" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "2*300*RPi;\n2*300*Pi;\ndiff
erence = evalf( % - %%);" }}}{PARA 0 "" 0 "" {TEXT -1 263 "\nAbout 1/2
0 of a foot too much. Thats not bad at all. What if the wanted to comp
ute the distance the earth travels, using the \"facts\" that the sun t
ravels in a circular orbit about the sun (actually its elliptical) and
 the radius is 10,000 miles (grossly wrong).\n" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "2*10000*RPi;\n2*10000*Pi;\ndifference = evalf( %
 - %%);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "\n \n\n" }{TEXT 269 3 "   " }
{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 274 19 " = sqrt(98694)/100?" }
{TEXT -1 63 "\n\nThe Hindus used this approximation. Lets see how it f
ares.\n  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "HPi := sqrt(986
94)/100;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "HPi/Pi;   % = e
valf(%,10); \n100*(1-evalf(HPi/Pi,10));" }}}{PARA 0 "" 0 "" {TEXT -1 
142 "\n\nThis approximation is differs by about 1/1000 of 1 percent! I
f the architect used this value on his colliseum, how much would he be
 off by?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "2*300*HPi;\n2*
300*Pi;\ndifference = evalf( % - %%);" }}}{PARA 0 "" 0 "" {TEXT -1 
240 "\nAbout 1/5 of a foot too much. What if the wanted to compute the
 distance the earth travels, using the \"facts\" that the sun travels \+
in a circular orbit about the sun (actually its elliptical) and the ra
dius is 10,000 miles (grossly wrong).\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "2*10000*HPi;\n2*10000*Pi;\ndifference = evalf( % - %%
);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 1 " " }{TEXT 267 38 "4. Other Fractional Approximations to " 
}{XPPEDIT 18 0 "Pi" "6#%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 59 "\nThere a
re even more accurate fractional approximations to " }{XPPEDIT 18 0 "P
i" "6#%#PiG" }{TEXT -1 60 ". Lets look at a series of increasingly acc
urate fractions.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "FracA
pprox := 22/7;\nx := evalf( %);\n`Percentage Error` = 100* abs( evalf(
 (Pi-evalf(FracApprox))/Pi, 50) ); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 111 "FracApprox := 333/106;\nx := evalf( %);\n`Percentage
 Error` = 100* abs( evalf( (Pi-evalf(FracApprox))/Pi, 50) ); " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "FracApprox := 355/113;\nx :
= evalf( %);\n`Percentage Error` = 100* abs( evalf( (Pi-evalf(FracAppr
ox))/Pi, 50) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "FracApp
rox := 103993/33102;\nx := evalf( %);\n`Percentage Error` = 100* abs( \+
evalf( (Pi-evalf(FracApprox))/Pi, 50) );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "FracApprox := 104348/33215;\nx := evalf( %);\n`Perce
ntage Error` = 100* abs( evalf( (Pi-evalf(FracApprox))/Pi, 50) );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "FracApprox := 208341/66317;
\nx := evalf( %);\n`Percentage Error` = 100* abs( evalf( (Pi-evalf(Fra
cApprox))/Pi, 50) ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "F
racApprox := 312689/99532;\nx := evalf( %);\n`Percentage Error` = 100*
 abs( evalf( (Pi-evalf(FracApprox))/Pi, 50) );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 117 "FracApprox := 833719/265381;\nx := evalf( %);
\n`Percentage Error` = 100* abs( evalf( (Pi-evalf(FracApprox))/Pi, 50)
 ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "FracApprox := 1146
408/364913;\nx := evalf( %);\n`Percentage Error` = 100* abs( evalf( (P
i-evalf(FracApprox))/Pi, 50) ); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 119 "FracApprox := 4272943/1360120;\nx := evalf( %);\n`Pe
rcentage Error` = 100* abs( evalf( (Pi-evalf(FracApprox))/Pi, 50) ); \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 68 "\nSo using \"lowly\" fractions, we can get some pretty go
od accuracy!\n " }}}{PARA 0 "" 0 "" {TEXT 259 36 "\n         \251 2002
 Waterloo Maple Inc " }}}{MARK "0 1" 47 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }