Big numbers

Implement big numbers! Start with non-negative numbers (also called natural numbers). We can implement a natural number as a list of Int, each element in the list can be seen as a digit in a base. Define a global constant
    base :: Int
which is the base for the digits. (So if the base is 10 then we get the ususal decimal representation). It will be convenient to store the numbers in reverse order, i.e. the decimal number 12569 will be represented as the list [9,6,5,2,1].

Implement addition and multiplication! You should use the same algorithm as the one you learned in school.

Now we can implement big integers by using the following definition:
   data Biginteger = Positive [Int] | Negative [Int]
Define then the standard functions for addition and multiplication.

If you want, you can use the following function to produce very big numbers:

   R  0     m  n    = mul m n
   R  (k+1) m  0    = 1
   R  (k+1) m (n+1) = R k m (R (k+1) m n)

If you look at this function carefully, you will notice that R 0 is iterated addition (i.e. multiplication), R 1 is iterated multiplication (i.e. exponentiation), R 2 is iterated exponentiation (i.e. tower) and so on. This function grows exceptionally fast in its first argument. For instance, already R 3 3 3 is a number which not only is bigger than the number of atoms in the universe, it cannot even be written down, using one atom for each digit in the number. I don't even want to think about the value of R ( R 3 3 3 ) ( R 3 3 3 ) ( R 3 3 3 ) !

If you want to learn more about this function, you can study pages 11 and onwards in the document Primitive recursive functions where the function is denoted by a double arrow.

Your solution to the homework should contain some examples and all the relevant definitions. It should also contain answers to the following questions:

  1. What is the maximal value of the base for your algorithm to work? Why?
  2. Does it work if the base is 1? What happens if the base is too small or too big?
  3. Why is it convenient to store the digits in reverse order?
  4. Are you able to get the following result for 1000! ? Don't worry if you cannot, I did not try it.
    40238726007709377354370243392300398571937486421071 46325437999104299385123986290205920442084869694048 00479988610197196058631666872994808558901323829669 94459099742450408707375991882362772718873251977950 59509952761208749754624970436014182780946464962910 56393887437886487337119181045825783647849977012476 63288983595573543251318532395846307555740911426241 74743493475534286465766116677973966688202912073791 43853719588249808126867838374559731746136085379534 52422158659320192809087829730843139284440328123155 86110369768013573042161687476096758713483120254785 89320767169132448426236131412508780208000261683151 02734182797770478463586817016436502415369139828126 48102130927612448963599287051149649754199093422215 66832572080821333186116811553615836546984046708975 60290095053761647584772842188967964624494516076535 34081989013854424879849599533191017233555566021394 50399736280750137837615307127761926849034352625200 01588853514733161170210396817592151090778801939317 81141945452572238655414610628921879602238389714760 88506276862967146674697562911234082439208160153780 88989396451826324367161676217916890977991190375403 12746222899880051954444142820121873617459926429565 81746628302955570299024324153181617210465832036786 90611726015878352075151628422554026517048330422614 39742869330616908979684825901254583271682264580665 26769958652682272807075781391858178889652208164348 34482599326604336766017699961283186078838615027946 59551311565520360939881806121385586003014356945272 24206344631797460594682573103790084024432438465657 24501440282188525247093519062092902313649327349756 55139587205596542287497740114133469627154228458623 77387538230483865688976461927383814900140767310446 64025989949022222176590433990188601856652648506179 97023561938970178600408118897299183110211712298459 01641921068884387121855646124960798722908519296819 37238864261483965738229112312502418664935314397013 74285319266498753372189406942814341185201580141233 44828015051399694290153483077644569099073152433278 28826986460278986432113908350621709500259738986355 42771967428222487575867657523442202075736305694988 25087968928162753848863396909959826280956121450994 87170124451646126037902930912088908694202851064018 21543994571568059418727489980942547421735824010636 77404595741785160829230135358081840096996372524230 56085590370062427124341690900415369010593398383577 79394109700277534720000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 000000000000000000.
  5. Finally, I want you to present yourself. Tell me your interests in computer science and what kind of research you are doing (or intending to do). A paragraph or two is enough!

Last modified: Thu Oct 9 10:46:39 KST 2003