My point is that rationals aren't correct in the way you're describing. For example, try rotating that rational ill-conditioned matrix. You'll probably have to

My point is that rationals aren't correct in the way you're describing. For example, try rotating that rational ill-conditioned matrix. You'll probably have to pick a precision to approximate the result and at that point you would have been better off with arbitrary precision floating point, algebraic or symbolic representations instead. None of these representations are wrong, they're simply tradeoffs for different usages. That's why IEEE floats are such a good default. They balance those tradeoffs extremely well, with fixed semantics and near-universal hardware support.