> Another way to look at what irrational numbers are is to say that they sort of don't really exist, they are like limits, or some ideals that cannot be reached because you'd need to spend infinity to reach that exact number when counting, measuring etc.
Depending on your definition of "existence", rational numbers (or any numbers) don't exist either.
I think it's kind of obvious what my definition of existence could be from the answer above: if it's possible to count up to that number in finite time, that number exists. By counting I mean a physical process that requires discrete non-zero intervals between counts. And you don't have to count in integers, you can count in fractions, not necessarily equal at each step: the only requirement is that the element used for counting exists (in terms of this definition) and that you are able to accomplish counting in finite time.
To me, this pretty much captures what people understand the numbers to be used for outside of college math (so no transfinite, cardinals etc.)
Do you mean the computable numbers? (there's an algorithm to compute them to arbitrary precision)
The irrational numbers used outside of college math, like pi or e or sqrt(2), are computable, though almost all are not.
You can do a lot of productive math using just computable numbers since they form a real closed field [1]. I believe they're a little harder to work with though.
Computable numbers are those that are described by computable functions. Irrationals like pi can be described by computable functions that take a precision as input.
Yeah... the article has a lot of "simplifications" that the reader has to just kind of trust the author on... and those are meant for people with not a lot of mathematical sophistication. All this talk about "shrink fast enough" and why it's important is just some intense handwaving w/o any actual explanation.
To be fair, it's kind of upsetting, but maybe there's no way to help it... some mathematical proofs can be "dumbed down" to the point that people with very little background can understand them. The proof of sqrt(2) being irrational might be one of those. But, what's given in the article feels like either the author didn't really understand the subject, or she couldn't explain what she understood in simple terms. But, it's really rare that there's such an easy to understand proof or concept. So, I don't blame her.
Depending on your definition of "existence", rational numbers (or any numbers) don't exist either.