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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 86 "Computer a
lgebra, week 1, lecture 1:\nAssignments in Maple, sequences, lists, an
d sets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
60 "Assigning a value to a variable in Maple can be done with :=" }}
{PARA 0 "" 0 "" {TEXT -1 101 "An assignment is a Maple command, and an
y Maple command ends with a colon or a semicolon, with : or ;" }}
{PARA 0 "" 0 "" {TEXT -1 108 "(the only difference is that if you end \+
a command with a colon, then nothing will be printed to the screen)." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Example
s of assignments: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A:=3;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B:=4;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 102 "To view the value of a variable in Maple, just type
 the name of the variable, and end with a semicolon" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 2 "B;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "You can assign two (
or more) variables at the same time:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "(A, B) := (5, 7);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 56 "Spaces have no meaning, so this command was the same as:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "(A,B):=    (   5,7)   ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 2 "B;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Simulta
neously assigning several variables at the same time is convenient if \+
you want to switch the values of A and B:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "(A,B) := (B,A);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "B;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Variables can also be used even w
hen they have not yet been assigned a value:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 2 "x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "y
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "P := x^3+x^5+12*x;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "When x is not assigned, you can s
till use x. Here P is a " }{TEXT 258 11 "polynomial " }{TEXT -1 125 "i
n x. If x had a value, say, an integer, then P wouldn't have become a \+
polynomial but it would have been an integer. You can " }{TEXT 269 4 "
eval" }{TEXT -1 33 "uate P at x = 1, 2, 3 as follows:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(P, x=1);  " }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 71 "The above command substituted x=1 into P and then
 evaluated the result." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e
val(P, x=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(P, x=3
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "or with the " }{TEXT 270 4 
"subs" }{TEXT -1 177 " command (subs is short for substitution, and do
es pretty much the same as the eval command.  With subs, the x=.. come
s at the beginning, and with eval, the x=.. is at the end)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "subs(x=3, P); # This command
 substitutes x=1 into P:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Comm
ents can be added to a line of Maple commands by using the symbol #  (
Maple simply ignores everything that comes after a #)." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "We could do a seque
nce of evaluations by the command seq (stands for sequence) as follows
. Lets say we want to substitute i=1..6 for x:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "seq( eval(P, x=i) ,i=1..6);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "seq(i, i=1..6);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 129 "The output of the seq command is a datastructure that'
s called a sequence in Maple. In Maple a sequence is something of the \+
form:" }}{PARA 0 "" 0 "" {TEXT -1 42 "   expression, expression, ..., \+
expression" }}{PARA 0 "" 0 "" {TEXT -1 3 "So " }{TEXT 259 58 "a sequen
ce is a number of expressions seperated by commas." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "Instead of computing the
 sequence eval(P, x=1), eval(P, x=2), ..  we can also compute the sum:
 eval(P, x=1)+eval(P, x=2)+... with the add command:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "add( eval(P, x=i), i=1..6);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "add(i, i=1..100);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 15 "Multiplication:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "mul(x-i, i=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "mul(i, i=1..30);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "30! ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "if
actor(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The command " }
{TEXT 260 7 "ifactor" }{TEXT -1 12 " stands for " }{TEXT 261 21 "integ
er factorization" }{TEXT -1 266 ". Every Maple command must be ended w
ith ; or with : and until now we've only used ;     (you only end a co
mmand with a colon : if you don't want to see the output of that comma
nd. This is useful if the output is very large, filling up too much sp
ace on the screen)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A:=2;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "View the contents of variable
 A as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Now end the command with a colon i
nstead of a semi-colon" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A
:=1234^1234:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "You see that wit
h a colon, Maple prints nothing. However, the command (in this case, a
n assignment) did get executed, see:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "In Maple, e
very command is followed by a colon :  or by a semi-colon ;" }}{PARA 
0 "" 0 "" {TEXT -1 154 "If you use : or ; that makes no difference for
 what Maple does, the only difference is that ; causes something to be
 written to the screen and : does not." }}{PARA 0 "" 0 "" {TEXT -1 85 
"Lets look some more at sequences in Maple. The following is an exampl
e of a sequence:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A := 3,
4,5,2,3,3,3,1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Now you can ext
end the sequence to the left, or to the right, or both, as follows:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "A:=A,20,30;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A:=hello,how,are,you,A,1000;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A[1];" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "A[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "A[3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "There is one speci
al sequence in Maple, namely the empty sequence, denoted by NULL. See \+
what happens:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A:=NULL,NU
LL,A,NULL;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Maple sequences are
 very handy. There is one problem with sequences though, and that is:
" }}{PARA 256 "" 0 "" {TEXT -1 38 "You can't tell the difference betwe
en:" }}{PARA 257 "" 0 "" {TEXT -1 21 " 1)  An expression a." }}{PARA 
258 "" 0 "" {TEXT -1 38 " 2)  A sequence with just 1 element a." }}
{PARA 0 "" 0 "" {TEXT -1 111 "Because of this problem, it is often mor
e convenient to work with lists in Maple than with sequences. In Maple
," }}{PARA 0 "" 0 "" {TEXT 257 48 "   a list is a sequence between squ
are brackets." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We can " }
{TEXT 262 100 "turn a sequence into a list by putting square brackets \+
around the sequence and using an assignment, " }{TEXT -1 11 "as follow
s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A:=[A];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 130 "We can again look at individual elements
, A[1], A[2] and such. The following command tells you the number of e
lements in the list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops
(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "A[15];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "A[nops(A)];" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 170 "To get the last element, you can do A[nops(A)];  A \+
shorter syntax, which does exactly the same, is A[-1], which gives you
 the last element. A[-2] is the second last, etc." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "A[-1]; # last element" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "A[-2]; # second last element" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "A[-3]; # third last element, etc." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "As already mentioned, everything \+
after # are considered comments (so everything after # will be ignored
 in Maple computations)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 263 78 "The seq command is very useful for operations wit
h: sequences, lists and sets." }}{PARA 0 "" 0 "" {TEXT -1 88 "If we wa
nt to let i be all elements of a list or a set A we can do:\n  seq( ..
..  , i=A);" }}{PARA 0 "" 0 "" {TEXT -1 71 "and if we want to let i be
 all elements of a sequence S then we can do:" }}{PARA 0 "" 0 "" 
{TEXT -1 98 "  seq( ....  , i=[S]);\nFor example, compute the sequence
 of squares of the elements of the list A:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "seq(i^2, i=A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
23 "Or the list of squares:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "[seq(i^2, i=A)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Or the se
t of squares." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{TEXT 264 55 "A set \+
in Maple is a sequence between curly brackets \{\}." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "Sets differ from lists
 in two ways: First, no element appears more than once (doubles will b
e automatically removed, something that often comes in handy). Second,
 the ordering has no meaning for sets." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 253 "If S is a non-empty set, then S[1] i
s an element of that set but you don't know in advance which element y
ou will get (if you try it twice, then S[1] will be the same both time
s, but your neighbor typing the same commands might get a different el
ement)." }}{PARA 0 "" 0 "" {TEXT -1 196 "This is the reason that redoi
ng the same computation in Maple can lead to a different (but mathemat
ically equivalent) result.  Say for example you solve a system of line
ar equations (with Maple's " }{TEXT 271 5 "solve" }{TEXT -1 140 " comm
and), and that the end result you get is \{x = 1, y = -z\}. Your neigh
bor might find \{x = 1, z = -y\} even if she types the same commands.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "\{seq(i^2, i=A)\};" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "We can use the A[-1], A[-2], const
ruction to reverse the list:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 29 "[ seq(A[-i], i=1..nops(A)) ];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 258 "We have seen how you can convert a sequence to a list, with [ \+
], or to a set, with \{\}. The converse is also possible. You can remo
ve the [ ] or the \{ \} brackets by the op command. This is also usefu
l if you want to convert a list to a set or a set to a list." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "B:=op(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "C:=\{op(A)\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "nops
(A), nops(C); # C has fewer elements because multiple elements are rem
oved." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "nops(B); # The nop
s command allows a list or set but not a sequence:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 62 "nops([B]); # Now the input of nops is [B] w
hich is a list.    " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 444 "As you ca
n see, we converted list A to a sequence B, and to a set C. Since C ha
d fewer elements than A, we can see that A must have elements that app
ear more than once. Note that nops does not work for a sequence, but i
t does work for lists and sets. So, to count the number of elements in
 a sequence, we have to put brackets [ ] around it before we can count
 with nops. Given a list, we can compute sublists as follows, using th
e double dot .." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "A[1..1];
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "A[1..4];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "A[1..-1];" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 10 "A[-3..-1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
232 "The -1'th element refers to the last elements. So elements number
s 1..-1 gives all elements.  Likewise, the -3'rd element is the third \+
last elements. So A[-3..-1] is the third last to the last element (i.e
. the last three elements)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "A[13..15];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "A[13..-1]
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "A[-3..15];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "C;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "C[1..4];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"C minus C[1..4];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The commands
 minus and union can only be used for sets." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 2 "B;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "B[4
..6];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 "We have seen before tha
t if you have an expression involving a variable x, you can replace x \+
by something else with an eval command (or a subs command). How to do \+
that for sequences?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "P;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=0,P);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=1,P);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "subsop(1=Hi,A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Note that this subsop command d
oesn't actually change A, it just returns A with its first entry repla
ced by Hi." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{TEXT 265 39 "To change
 A we have to do an assignment" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "A := subsop(1=Hi,A);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "A := subsop(2=there, A);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "A := subsop(3=fox, A);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "A:=subsop(4=NULL, A);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "nops(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "
Here I replaced entry A[4] by the empty sequence NULL, so this is the \+
way " }{TEXT 266 35 "you can remove an entry from a list" }{TEXT -1 
45 ". For sets there are different ways to do it." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 2 "C;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "subs(hello=NULL,C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "
C;  # the previous line didn't change C because there was no :=" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "C minus \{hello\};" }}}
{EXCHG {PARA 259 "" 0 "" {TEXT -1 61 "The best way to remove an elemen
t e from a set S is by doing:" }}{PARA 260 "" 0 "" {TEXT -1 19 "  S :=
 S minus \{e\};" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 118 "Note that if A,B are sets then in \"A minus B\" the set \+
B can be any set, it does not need to be a subset of A. Example:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "\{hello,Hello\} minus C;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "In Maple, the variable Hello is \+
different from hello because lower case and upper case is not the same
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 25 "Ano
ther example with sets" }{TEXT -1 125 ": Compute all numbers that are \+
a square of a positive integer <= 100, and that are also a sum of two \+
squares. How to do that?" }}{PARA 0 "" 0 "" {TEXT -1 41 "Well, first t
ake all squares of i=1..100." }}{PARA 0 "" 0 "" {TEXT -1 48 "Then let \+
i = all elements of that set of squares" }}{PARA 0 "" 0 "" {TEXT -1 
48 "and let j = all elements of that set of squares," }}{PARA 0 "" 0 "
" {TEXT -1 115 "and for all of those, take i+j, and then take the set \+
of all of that.\nThen take the intersection of these two sets." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "squares:=\{seq(i^2,i=1..100)
\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "sum_squares := \{ se
q(seq(  i+j  ,i=squares),j=squares) \}:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 200 "I used a colon instead of semi-colon to end the command,
 to prevent Maple from printing lots of numbers. To find the answer to
 the question, we will have to compute the intersection of those two s
ets:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "squares intersect s
um_squares;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "In case the resul
t is not sorted (it need not be, because sets don't have an order) we \+
can sort it by first turning it into a list, and then sorting:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "L := [op(squares intersect s
um_squares)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "L := sort(
 L );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 56 "Recalling previous expr
essions with the percentage sign." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 154 "The percentage sign % in Maple refers to
 the last thing Maple evaluated. And %% means the one before that, and
 %%% the one before that (there is no %%%%)." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 7 "a := 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "b:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=3;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "d:=4;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "%%;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "At \+
this point, the last one is 3, and so the second last is no longer 3, \+
it's 4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "%%;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "b;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "c;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "%%%;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "%%%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
4 "%%%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "b;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "%;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "%%%;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 121 "If you end a command with a colon, it still counts \+
as an expression, even if you don't see it on the screen. For example,
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int( ln(x), x):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Now the last thing evaluated will \+
be the " }{TEXT 272 3 "int" }{TEXT -1 121 "egral of ln(x), so if I typ
e % the result won't be 2 (the last thing visible on the screen) but i
t will be that integral:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "
%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 6 0" 
25 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }