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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 399 "Number system with base b
eta. Let beta be some fixed positive integer > 1, and let [a.n,a.(n-1)
,....,a0] denote the integer a.n * beta^n +... + a.0 * beta^0 where ea
ch a.i is an integer in the range 0 .. beta-1. The a.i are called digi
ts. They are the usual digits 0..9 when beta=10, which is the common w
ay humans represent numbers. They are 0 or 1 when beta=2, which is com
mon inside the computer." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "
beta:=10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "L:=[1,2,3,4,5,
6,5,4,3,2,1,0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "list_to_
integer:=proc(L,beta)\n   local i;\n   add(L[-i]*beta^(i-1), i=1..nops
(L))\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "list_to_integ
er(L,10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 274 "Computers can store
 lists of arbitrary (depending on the amount of memory) lenght. This w
ay we can store integers with arbitrary many digits. In this way of de
noting integers beta is called the base of the number system. If we ta
ke beta=1000 then each entry stores 3 digits." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "beta:=1000;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "L:=[1,2,1,2,345,678];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "list_to_integer(L,beta);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 81 "Computers often store numbers internally with a base beta
=2 or beta=a power of 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
beta:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "L:=[1,1,1,0,0,0
,0,1,1,1,1,1,0,0,1,0,1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 
"list_to_integer(L,beta);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Let
s write some procedures to add and multiply numbers represented in bas
e beta, to get an idea how a computer could multiply large numbers." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 361 "add_lists:=proc(L1,L2,bet
a)\n   local n,res,i,q,r;\n   n:=nops(L1)-nops(L2);\n   if n<0 then\n \+
      RETURN( procname(L2,L1,beta) )\n   elif n>0 then\n       RETURN(
 procname(L1,[0$n,op(L2)],beta) )\n   fi;\n   n:=nops(L1);\n   res:=NU
LL;\n   q:=0;\n   for i from 1 to n do\n      res := irem(q+L1[-i]+L2[
-i],beta,'q'),res\n   od;\n   if q>0 then res:=q,res fi;\n   [res]\nen
d;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "beta:=10;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L1, L2 := [7,2], [5,5];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "add_lists(L1,L2,beta);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "add_lists(%,L2,beta);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "add_lists(%,L2,beta);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "add_lists(L1,%,beta);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "mult_lists:=proc(L1,L2,beta)\n  lo
cal i,j,res;\n  res:=[0];\n  for i from 1 to nops(L1) do\n      res:=a
dd_lists(res,[seq(j*L1[-i],j=L2),0$(i-1)],beta)\n  od;\n  res\nend;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "mult_lists([1,2,3,4],[2,3,
4],10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "1234*234;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "beta:=1000;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "L1:=[seq(rand(0..beta-1)(),i=1..100
)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "L2:=[seq(rand(0..bet
a-1)(),i=1..100)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "time(
);  mult_lists(L1,L2,beta); time();" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "29" 0 }{VIEWOPTS 1 1 0 1 1 1803 }