[fsmv]

I've also felt that real numbers aren't real ever since I tried doing algebra in floating point numbers and finding that some formulas that can be reversed in math cannot be reversed on a computer.

In the real numbers there exists a number that encodes the answer to the halting problem for every number, it's provable and also provable that you cannot write a program to print the digits.

Of course it's a computer scientist doing this. I'll have to read his papers. I wonder if he has anything about rational differential equations.

[fjfaase]

Floating numbers with a fixed precision based on a fixed base, are really limited on what rational numbers can be represented. For example, the standard floating point formats cannot exactly represent the fraction 1/3. One would expect that 3 times 1/3 would result in 1, but it does not. This is based on an implementation decision on how to represent rational numbers. There are computer algebraic systems that represent fractions like 1/3 exactly, such that 1 times 1/3 is equal to 1. In a similar way it is possible to represent certain real numbers like the square root of 2, such that the square of such a number is exactly equal to 2. To being able to print such a number in a certain base (10 for example) does not mean it is not possible to represent such numbers in a way that it is possible to perform calculations with these.

One should realize that the implementation choice to represent floating numbers in the way it is done in processors is related to performance and memory usage.