Sure,there are a large number of "computable reals" such as square roots and taylor series expansions (pi, e, sine, cosine, etc.) which can be estimated to grea

Sure,there are a large number of "computable reals" such as square roots and taylor series expansions (pi, e, sine, cosine, etc.) which can be estimated to greater and greater precision (always with rational numbers) so we can reasonably talk about them in the abstract (such as use them in equations) much in the same way we talk about the imaginary numbers, quarternions, and octonions. But the idea that you need to construct an entire field of completely undefinable infinite chains of decimals just to allow for a small handful of irrational numbers is kind of absurd. Norman Wildberger (among others) has put a lot of effort into building up the foundations of mathematics using only the rational numbers and it is quite interesting how far he has gone. He's considered to be a crank by many, but you have to appreciate his effort. To go back to your original question, his Math Foundations video series, especially #14, #95, and #96 might help.