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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 50 "A First Look at the Arithmetic \+
Properties of Maple" }}{PARA 257 "" 0 "" {TEXT -1 19 "Jim Herod, Retir
ed " }}{PARA 258 "" 0 "" {TEXT -1 22 "School of Mathematics " }}{PARA
259 "" 0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 260 "" 0 "" {TEXT -1
21 "herod@math.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 13 " Computer" }{TEXT 256 236 " Algebra Systems ha
ve changed the way we think about doing mathematics in our work and in
our teaching. All the computer algebra systems must be good at the ar
ithmetic for symbols, as well as for numbers. \n\n Here is a first
look at " }{TEXT 257 19 "Maple's arithmetic." }{TEXT 258 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "A
rithmetic With Numbers" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT 259 38 "First, there is the simple arithmetic:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2+5;"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "(2+3)^2*7/5;" }}}{PARA 0
"" 0 "" {TEXT -1 10 "There are " }{TEXT 260 43 "some numbers that Mapl
e knows symbolically." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sin
(Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(sin(Pi/4))
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(Pi,4);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "sin(%/4);" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Ther
e are other built-in constants." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 18 "evalf(gamma);\nI^2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 10 "You could " }{TEXT 261 215 "have guessed almost
all of the above output. What about the following? Humans know that t
he result of the first computation below is zero. Be aware that Maple \+
may not make that simplification if it is not requested." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "sqrt(999983) - (999983^(1/6))^3;\ns
qrt(999983) - (999983^3)^(1/6);\nsimplify(%,radical);" }}}{PARA 0 ""
0 "" {TEXT -1 36 "Some simplifications are a surprise:" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(sqrt(2*sqrt(3)+4));" }}}
{PARA 0 "" 0 "" {TEXT -1 17 "Other's are hard:" }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 37 "sqrt(2*(3-sqrt(2)-sqrt(3)+sqrt(6))); " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(%,radical);" }}}
{PARA 0 "" 0 "" {TEXT -1 40 "Sometimes, more help must be brought in:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " radnormal(sqrt(2*(3-sqr
t(2)-sqrt(3)+sqrt(6))));" }}}{PARA 0 "" 0 "" {TEXT -1 84 "One of the a
ttractive things about a good Computer Algebra System is that it can d
o " }{TEXT 262 5 "exact" }{TEXT -1 39 " integer arithmetic. Compare th
ese two." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "(2/5)^32;\n.4^32
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(%%);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 100 "W
e have illustrated the ability of Maple to do numercial arithmetic. Wh
at about symbolic arithmetic?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Exercise for the reader" }}{PARA
0 "" 0 "" {TEXT -1 8 "Compute " }{XPPEDIT 18 0 "sqrt(Pi);" "6#-%%sqrtG
6#%#PiG" }{TEXT -1 31 " accurate to 20 decimal places." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1
23 "Arithmetic with Symbols" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 173 " The point of the computer alegbra system \+
is to do symbolic arithmetic, not just numerical arithmetic. For examp
le, the following expression is not the square root of y." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "sqrt(
x*y*abs(z)^2);\nsqrt(x)*abs(z);\nsqrt(x*y*abs(z)^2)/(sqrt(x)*abs(z));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify(%,radical);\nr
adnormal(%);" }}}{PARA 0 "" 0 "" {TEXT -1 12 "To see that " }{XPPEDIT
18 0 "sqrt(x*y)/sqrt(x);" "6#*&-%%sqrtG6#*&%\"xG\"\"\"%\"yGF)F)-F%6#F(
!\"\"" }{TEXT -1 20 " is not necessarily " }{XPPEDIT 18 0 "sqrt(y);" "
6#-%%sqrtG6#%\"yG" }{TEXT -1 27 ", we provide illustrations." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "x:=-I; y:=-I;\nsqrt(x*y);\n
sqrt(x);\nsimplify(sqrt(x*y)/sqrt(x));\nsqrt(y);\n\n" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "What is \+
it we were thinking when we thought " }{XPPEDIT 18 0 "sqrt(x*y)/sqrt(x
);" "6#*&-%%sqrtG6#*&%\"xG\"\"\"%\"yGF)F)-F%6#F(!\"\"" }{TEXT -1 3 " =
" }{XPPEDIT 18 0 "sqrt(y);" "6#-%%sqrtG6#%\"yG" }{TEXT -1 47 "? We w
ere thinking that x and y are positive. " }}{PARA 0 "" 0 "" {TEXT -1
83 " We set x and y back to symbolic constants and then choose the
m to be positive." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x:='x';
\ny:='y';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "assume(x>0);\n
assume(y>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(sq
rt(x*y)/sqrt(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x:='x'
;\ny:='y';" }}}{PARA 0 "" 0 "" {TEXT -1 44 "This last example was not \+
so different from " }{XPPEDIT 18 0 "sqrt(a^2);" "6#-%%sqrtG6#*$%\"aG\"
\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sqrt(
a^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "assume(a>0): sqrt(
a^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "assume(a<0): sqrt(
a^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a:='a':" }}}{PARA
0 "" 0 "" {TEXT -1 33 "Here is more symbolic arithmetic." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "(x+y)^3;\nexpand((x+y)^3);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "coeff((x+y)^3,x,2);" }}}
{PARA 0 "" 0 "" {TEXT -1 87 "As a connection between numerical and sym
bolic arithmetic, consider these calculations:" }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 14 "sum(p,p=1..5);" }}}{PARA 0 "" 0 "" {TEXT -1 34 "
Here is a more daring calculation." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 14 "sum(p,p=1..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 20 "factor(simplify(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Exercise for the reader
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Calcu
late " }{XPPEDIT 18 0 "sum(1/(p^2),p = 1 .. infinity);" "6#-%$sumG6$*&
\"\"\"F'*$%\"pG\"\"#!\"\"/F);F'%)infinityG" }{TEXT -1 1 "." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "A
rithmetic with inequalities" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 112 "Maple can do arithmetic with inequalities. A f
irst simple inequality is to ask what for what x's is it true that" }}
{PARA 0 "" 0 "" {TEXT -1 102 " 3 x - 5 < 4.\nPerhaps the \+
only thing interesting about the answer is how it is expressed." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(3*x-5 < 4,x);" }}}
{PARA 0 "" 0 "" {TEXT -1 6 "Maple " }{TEXT 265 171 "gives an opportuni
ty to visualize inequalities. The previous asked that the graph of \n
\011\011\011\011\011\011 y = 3 x - 5 \nshou
ld lie below the graph of the cons" }{TEXT -1 0 "" }{TEXT 263 4 "tant
" }{TEXT 264 82 " 4. The response was this happens for x < 3. It would
be useful to visualize this." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 24 "plot(\{3*x-5,4\},x=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 235 "Thi
s graph provides a visualization of the solution to the inequality. To
approximate the answer, use the mouse to touch the intersection to th
e two lines and look in the left-hand corner to see the x-coordinate (
and the y-coordinate)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 42 " A problem not conceptually harder \+
is " }{TEXT 266 3 "the" }{TEXT -1 1 " " }{TEXT 267 20 " following: sup
pose " }{TEXT 268 23 "\n\011 \011\011" }{TEXT 269
32 "y = 3 x - 12 and z = - 2 x + 9" }{TEXT 270 2 ".\n" }{TEXT 271
68 "For what x's is y < z? The answer is the solution to the inequali
ty" }{TEXT 272 37 "\n\011\011\011 " }
{TEXT 273 21 "3 x - 12 < - 2 x + 9" }{TEXT 274 1 " " }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 28 "solve(3*x-12 < -2*x + 9, x);" }}}{PARA 0
"" 0 "" {TEXT 275 62 "Visually, the result comes with the intersection
of two lines." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 31 "plot(\{3*x-12, -2*x+9\},x=-5..6);" }}}{PARA 0 "" 0 "" {TEXT -1
75 "Use the mouse to approximate the maximum x for which this inequali
ty holds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
8 " Of " }{TEXT 276 74 "course, all inequalities are not linear. W
e ask where is (x+1)/(x+2) < 3. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 25 "solve((x-1)/(x-3) < 3,x);" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Now,
" }{TEXT 277 260 "the visualization is a little more interesting. We \+
draw the graph of the left-side and ask where the graph lies below the
line y = 3. Because we are uncomfortable with the graph going off to \+
infinity near x = 3, we restrict the range of y, as well as the range
" }{TEXT -1 6 " of x." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plo
t([(x-1)/(x-3),3],x=-2..5,y = -5..5,discont=true,color=[BLUE,RED]);" }
}}{PARA 0 "" 0 "" {TEXT -1 15 " You begin " }{TEXT 278 190 "to won
der how good the inequality arithmetic is. Let's make up a problem for
Maple. We generate a polynomial that has four crossings of the x-axis
and ask Maple to tell where it is negative." }}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 36 "p4:=expand((x+2)*(x+1)*(x-1)*(x-2));" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(p4 < 0, x);" }}}{PARA 0 "" 0
"" {TEXT -1 9 "Embolden " }{TEXT 279 29 "by this success, try another.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve((x^2+3*x+1)/(x^2-3
*x+2) < 1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 9 "It seems " }{TEXT 280
344 "that a graphical understanding of this last will give much better
intuition. Again, the possibility that the function blows up around x
= 1 or x = 2 causes us to restrict the y's. How much to restrict the \+
y's is a question that will arise when you see the graph, and is a que
stion that can be answered with results from the notions of calculus.
" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(\{(x
^2+3*x+1)/(x^2-3*x+2),1\},x=-3..3,y=-40..10);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 120 " One last inequality: For reasons that are cle
ar to a calculus teacher, the following inequalities are interesting:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "quotient:=(2*(x+h)^2+3*(
x+h)+5 - (2*x^2+3*x+5))/h;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
14 "epsilon:=.007;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "solve
(quotient-(4*x+3) " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Exercise for the \+
reader." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
29 " For what x's is - 2 < " }{TEXT -1 14 " 3 x + 7 < 5?" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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