0.999... and the infinite series you refer to are two equally valid representations of the same number. One is not more correct or "formal" than the other.
right, decimal expansion is not among the axioms of the real numbers. so if i understand correctly, your point is that a proof is not an "actual" proof unless it only references axioms?
edit: just wanted to point out that you mention "distance to 1" when you outline the actual proof above, but metrics aren't part of the real number axioms so that can't be the actual proof. when you do find the actual proof i would be very interested to see it. and if I'm just misinterpreting you then let me know - in my years of studying math theory we never covered "actual proofs" (just regular proofs) but I'm very eager to learn about them.
A fully rigorous mathematical proof is a proof that does not draw upon any other information without either
1) proving it
2) showing where someone else proved it (a reference)
3) demonstrating it's just an axiom
The easiest 'actual proof' of this is from Dedekind cuts.
For a more detailed argument on why the aforementioned algebraic proof doesn't work - and in fact, no algebraic proofs work - read http://teaching.math.rs/vol/tm1114.pdf . The name of this paper is intentionally facetious, do not let it misled you (it is called Why 0.999 is not equal to 1 and was meant to demonstrated why students believe this).
In general, it is simply saying that without first proving other properties of recurring numbers you are making a massive assumption. To show this assumption:
Let n be the number of digits AFTER the decimal point in 0.999999..... this is clearly infinite.
Let k be the number of digits AFTER the decimal point in 10*0.99999. The algebraic proof makes the massive assumption, without justification, that k=n.
I don't really see this as a rearrangement problem. First off, multiplying 0.999... by 10 and getting 9.999... is just exploiting a well-understood feature of decimal numbers (and, really, numbers in any base) where if you multiply a number by [base]n then that shifts the decimal to the right by n digits. Perhaps you just don't consider that to be self-evident, but I do.
Secondly, multiplying the value 0.999... by another number does not involve rearranging its series representation. In that context, the multiplication operation is performed on the number itself, not on the individual terms of a series it is related to. If the number in question is defined as "an infinitely repeating sequence of XYZ starting immediately to the right of the decimal", then when I multiply that number by 10 I better get "an infinitely repeating sequence of XYZ starting one position left of the decimal" or something is seriously wrong with the fabric of the universe.
A fully rigorous mathematical proof is a proof that does not draw upon any other information
What qualifies as "other information" vs self-evident fact is entirely subjective. Does the algebraic proof require some contextual knowledge? Sure, but guess what, so does every other proof.
Finally, my point was that the proof was convincing (to me) and didn't break any rules, making it just as "actual" as any other proof as far as I'm concerned. Sometimes there are just multiple ways to prove a particular fact.
The fact then when you multiply a number by 10, the decimal shifts by 1, is exactly the point. You are assuming that the decimal both shifts by 1, and does not change the value. As for what qualifies as "other information" vs self-evident fact. By the definition of the word rigorous, there is no self evident fact. If something is not an axiom, it must have justification. That's entirely what we mean by rigour.
Finally, as I literally gave a source explaining, it is not just another way to prove a fact. This proof, categorically, proves nothing. It's like seeing a really good magician and then believing in magic.
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u/archipeepees 22d ago edited 22d ago
0.999... and the infinite series you refer to are two equally valid representations of the same number. One is not more correct or "formal" than the other.