# Should there be a question on intuition for $0.999\dots = 1$?

I think that as described in this answer, Jonathan Fischoff burned a lot of reputation points asking questions to gain intuition on $$0.999\dots = 1$$ in MathOverflow. Even if that user really did lose a lot of points for trying to get intuition on $$0.999\dots = 1$$, just because they lost of lot of reputation points doesn't mean they didn't for real have an idea worthy of attention.

Some mathematicians have a demand for formalization and refuse to accept any theorem they don't know how to prove themself. I think there was

All of those questions ended up with a positive score. Maybe a question for intuition about $$0.999\dots = 1$$ would also be suitable.

For those of you who don't want to read all 6 of those questions I linked, you may want to read just some of the first 4 because they're more similar to the question I'm proposing which is a question about intuition for a counterintuitive result unlike the other 2 which are about a rigorous proof about an undisputed theorem.

• Life is short. Do you think you could get this down to 25 words or less? or at least post an "executive summary"? – Gerry Myerson Jul 1 at 5:47
• @GerryMyerson I think you're probably right that there was so much extra unnecessary detail so I guessed what it was an deleted it. – Timothy Jul 1 at 6:06
• Try making this point without having a dozen links to questions that you answered. Let's start there. – Asaf Karagila Jul 1 at 7:46
• We already have a question on this topic with 32 (non-deleted) answers and 99 linked questions, so I'm pretty sure that everything that could be said on this subject has been said multiple times already. – Arnaud D. Jul 1 at 15:46
• @ArnaudD. I read through all 27 currently existing answers. Some of them a long time before and some of them recently. Maybe some I didn't read through super carefully but I did do some reading of every single one of them. For me, it says 28 answers I guess because my own other deleted answer is visible for me. I think some people have a strong intuition that distinct decimal expansions represent different numbers and you can build a system based on that intuition dropping the criterion of a complete ordered field just like you can build a system based on normal people's intuition that R is a – Timothy Jul 2 at 19:33
• complete ordered field and that the meaning of a decimal expansion is a series. You could construct the dyadic rational numbers by continuously constructing numbers that are half of numbers you already constructed that are not yet divisible by 2 then construct the real numbers from those Dedekind cuts where the lower part has no maximal element nor does the higher part have a minimal element and show that $\sqrt{3}$ exists in that system. You also could construct the numbers that can be expressed as an integer plus an integral multiple of $\sqrt{3}$ as ordered pairs of integers where you – Timothy Jul 2 at 19:39
• define +, $\times$, and $\leq$ in the following way although these would be completely different objects from the ones that can be constructed that way in the other system. Addition is defined as addition of the individual components. Multiplication is defined by $(x, y) \times (z, w) = (xz + 3yw, xw + yz)$, and for inequality, $(z, w) > (x, y)$ if either $(z - x)^2 - 3(w - y)^2 > 0$ and $w - y \geq 0$; or $w - y < 0$ and either $(z - x)^2 - 3(w - y)^2 < 0$ or $(z - x) > 0$. We see that (2, 1) has a multiplicative inverse in this system and the square of the first component minus 3 times the – Timothy Jul 2 at 19:53
• square of the second component is preserved under multiplication by (2, 1). I made a mistake in how to define the ordering but I'm not sure I have time to change it before it's too late. The pair is meant to represent the first coordinate plus $\sqrt{3}$ times the second, not minus $\sqrt{3}$ times the second. Sometimes somebody finds a proof of a counterintuitive result more intuitive when you describe more of what's going on like I did in my answer to the second question I linked in this question and then got an upvote for. Maybe because some people find all answers to – Timothy Jul 2 at 20:02
• math.stackexchange.com/questions/11/… so unintuitive, it would be suitable to ask another question although maybe the only user I know might quite likely have that question might be Jonathan Fischoff whom I invited to participate in this discussion as I described in a comment on their answer to math.stackexchange.com/questions/11/… while being careful in how I wrote the comment to not lead to a destructive contribution to the network. I think Stack Exchange allows people to ask and answer their own questions – Timothy Jul 2 at 20:07
• because they can predict what question somebody else might have and never asked even though if they can answer it or what question somebody else might like reading and find so useful, it means they know the answer and don't have that question. I think that in this case, I cannot ask such a question myself because I already know the answer and others wouldn't find it so useful and the only person who can in good conscience ask it is somebody who actually has that question which might be Jonathan Fischoff. I think Jonathan Fischoff could formulate that question on the meta site first to figure – Timothy Jul 2 at 20:13
• @Timothy You're right about the number of answers, I miscounted the amount of deleted answers (I found it by searching "deleted by" and missed all the "deleted from review"). – Arnaud D. Jul 2 at 20:14
• out how it should be asked because as described in the 23rd comment under the question details of matheducators.stackexchange.com/questions/15475/…, a question can be formulated on the meta site. – Timothy Jul 2 at 20:15

I don't consider asking intuition (alone) for $$0.99\cdots=1$$ is appropriate. (For many people, the "intuition" is that they are not equal.)
A more fundamental (perhaps more useful) question(s) would be what is $$0.99\cdots$$ and what is $$1$$. Knowing exactly "what" helps greatly understanding "why" the statement is true.