[fpc-pascal] Super Large Integer Math Calculations

[noreply at z505.com]

For integers beyond 64 bit, or even beyond 32 bit on a 64 bit machine, 
why can't the math be broken down into peices the way a human does it on 
paper, and then theoretically any number can be added and subtracted, 
even if it is beyond 32/64 bit?

Example:

type TSuperLargeInt = string;

var
   i, j: TSuperLargeInt;
   output: TSuperLargeInt;
begin
   i := '100000000000000000009';
   j := '100000000000000000001';
   output := AddLargeInts(i,j);
   writeln(output);
end.

The function AddLargeInts simply breaks the math into peices and adds 9 
plus one, carries the one, then adds 1 plus 1, and writes the total as a 
string,, that the computer can output as a string, instead of an integer 
the cpu can handle directly. Strings allow concatenations of numbers to 
virtually any length (with the limit of the max string length, which is 
quite long) so when you use math to write numbers as strings, the number 
can be written any length you want.

And it doesn't have to be strings that are used, but some compiler magic 
that uses strings internally (concatenating numbers and writing 
numbers). Strings can be used as a prototype to prove the concept 
works... (I can provide a demo if needed).

Am I reinventing something that already exists somewhere in, the Math 
unit, or elsewhere, to have math operations on super large numbers 
beyond 32/64 bit?

This just covers addition/subtraction, which can be broken down into 
small pieces an int32 can easily handle (you are simply doing math from 
0-9 numbers, so its byte sized math!) - multiplication can likely be 
broken down into small pieces too but I haven't thought about it.

Surely someone has already thought of this and it exists somewhere?

To add the two large numbers above, simple write a function that:
  - adds the last number (9) and 1 in the string, and carries any 
remainder
  - adds the second last number, with the remainder, so result is 1
  - adds the third last...
  -  etc..
  - up until the first number (1)
  - write the output result as a concatenated/constructed string (not 
int32)

AFAICT any large number to ridiculous size limits can be added by 
breaking the math into small pieces that 32 bit numbers can handle (or 
rather byte sized numbers actually since you are doing 0-9 math 
addition).

This has got to be reinventing the wheel though: someone must have 
already invented this in Fortran or the math unit, or somewhere...?

And then the question becomes, what math cannot be done, by breaking 
down into small byte sized pieces.

Loops: not sure if these super large integers are any good in loops 
unless you loop an int32 around several times (compiler magic?).

The reason I use a simple case of adding 100000000000000000009 with 
another number is for simple test case that doesn't require much 
thought.. obviously 9 plus 1 means 10, so you carry the 1. then the rest 
is easy since you skip all the zeros and add the first 1 and 1 = 2. But 
the point is, addition on paper by humans is broken down to 0-9 
additions, so why not a CPU can do this too? Then output the result as a 
concatenated string even if the cpu cannot hold this number directly 
since it is bigger than int32.

Possibly I may be reinventing this:
http://www.delphiforfun.org/Programs/Library/big_integers.htm
Quote "All operations are performed pretty much the way you would do 
them with pencil and paper."

And:
http://rvelthuis.de/programs/bigintegers.html

A compiler could have magic inside it, or a unit could use operator 
overloading, to make it more built in and not use strings, but some 
custom type (maybe implemented as a string underneath, or array of 
bytes). Surely fortran or a number crunching language must have thought 
of this already, or the freepascal Math unit, or some BigInteger unit 
out there already.

Interesting Turing test: can all math be done up until infinity, on an 
int32 computer, if you just implement compiler magic to work around your 
int32 limitation, and break it down to byte sized (or int32 sized) 
pieces...