Rational arithmetic needs better matrix libraries. No loss of precision then.
First of all, the example in the article is computing eigenvalues. Let me ask you: how do you expect to calculate eigenvalues using rational numbers? As a refresher, if it's been years since you took a look at linear algebra, the eigenvalues of a matrix are the roots of the matrix's characteristic polynomial, and we remember from high-school algebra that the roots of a polynomial are often not rational numbers, even if the coefficients are rational. Therefore, using rational arithmetic here is a complete non-starter. It's just not even possible, right from the very start.
However, let's suppose that the particular problem that we are interested in solving does admit a rational solution. What then? Well, our real-world experience tells us that we often need an excessive amount computational resources and memory for exact, rational answers to linear algebra problems even of modest size.
As you perform more computations on rational numbers, the memory required continues to grow.
No no no no no no. Rational arithmetic satisfies (a+b)+c=a+(b+c); floating point arithmetic does not. Therefore floats are not a subset of rationals.
This reasoning is not sound. The conclusion is only correct by accident.
The reason floats are not a subset of the rationals is because floats distinguish positive and negative zero, and because floats represent non-finite values (NaN and infinity).
The fact that floating-point operations are not commutative is a consequence of the fact that floating-point numbers are not a "subring" of the rational numbers, not a consequence of the fact that the floating-point numbers are not a subset of the rationals. You could easily take a subset of the floating-point numbers that is also a subset of the rationals, it would still not be associative (assuming it contains at least two different elements).
Here is the concept you are looking for:
But it is worth pointing out that it is actually possible to maintain "perfect" accuracy when doing arithmetic even when sqrts (or other irrational numbers) are involved.
The idea is to store numbers as an AST consisting of an expression. Then you can perform any arithmetic on them by growing the AST. After each operation you simplify as much as you can. Once you have your final result you can evaluate the AST up to any precision that you need (and sometimes your final AST will already be a rational number because sqrts cancelled out)
In other cases, there are often better ways to figure out that the answer is correct. Figuring out that the answer is correct using rational arithmetic is often just too inefficient, compared to doing a bit of numerical analysis or figuring out a way to validate the answer you got. That is, unless the problem is very small.
And anything without a rational result you're stuck again.