Do irrational numbers actually exist?

It shows how bad the weather is, and has been, I am reduced to looking at number theory 🙂  I am not a Mathematician by any stretch of the imagination, so my basic question may already be well known, but I’ve come across an interesting problem when looking at infinity (I think lots of people have come across interesting problems when looking at infinity, including a good few who have gone bonkers while doing so – I think I can see why).

There is a beautiful simple little proof that shows that any number that has a repeating decimal sequence – NO MATTER HOW LONG THE REPEAT SEQUENCE – is a rational!  This immediately begs the question: “Is there any such thing as an irrational number?”  This depends entirely on the following: “Is there any proof that the irrationals do not repeat at infinity?”  There is of course the proof that irrationals (such as the square root of two) cannot be expressed as a rational fraction, so the rational fraction will be infinitely long.  Question is having got to the “end” of our infinitely long fraction, does it repeat?

Don’t think about this too much – it didn’t do people like Cantor much good 🙂

Update:  I have just been told by a famous Mathematician that “repeat at infinity” is meaningless.  Well that’s one good way out of the problem I guess 🙂

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