computer programmer's maths puzzle [closed]

A computer programmer looked at part of his code

x = x + 1;


and then thought what a strange equation that would be for a mathematician

$$x=x+1$$

The programmer asked some mathematicians if any number, $$x$$, would satisfy the mathematical version of the equation above.

Would it have been possible for one of the mathematicians to have suggested a number for $$x$$?

Note about edit I have tried to edit to give a well defined question - or at least better defined than the original question.

• Keep in mind that infinity is not a number – Adam Sep 27 '19 at 23:21
• @adam maybe that should be an answer.... – tom Sep 27 '19 at 23:29
• Possible duplicate of What number + 1 equals itself? – Belhenix Sep 28 '19 at 0:57
• Finally and most fatally, in my opinion, 3) how is this a puzzle? It seems more something to recognize than something to solve—that is, a trivia question more than a puzzle. Unless there is more to this than the answers to both questions have come remotely close to grasping, this isn’t a puzzle. (And if you just posed it to see if someone could find an answer that works, rather than posing it as a puzzle designed to hint toward a solvable answer, then I submit to you that you’ve posed that speculative question in the wrong site altogether.). – Rubio Sep 29 '19 at 12:48
• @tom You ... may not be able to delete, as it's already drawn answers. For some general advice, see my answer to What should I do if I've made a mistake in my question?, which covers this ground pretty well. For this question in particular, I was actually going to respond that even a flawed question can serve as a good example of what kind of things do not work well, so if nothing else, I think that's a fair disposition for it. Note too that I'm more than tempted to make the related question a duplicate of this one, which I may yet do. – Rubio Sep 29 '19 at 19:43

This seems to be a pretty open question to me. Some tries:

definition of a number

tell you about some of his thoughts about what qualifies as a number. If $$\pm$$ infinity qualifies as one, that would work for x.

Then of course there are

number spaces like $$\mod 1$$ in which e.g. $$0.5 \equiv 1.5 \mod 1$$

If you don't really have a

definition... I guess the mathematician would love to come up with something for you. For example just define that $$x$$ and $$1$$ should be "numbers", and $$+$$ means just ignoring all numbers after it. Then $$x = x + 1$$ would be true.

...but that's a pretty unusual thing to do. In real life I think I have already seen this:

$$1$$ meaning "everything" in set theory and $$+$$ meaning merging sets (being used instead of $$\cup$$). Then if $$x = 1$$ you merge two identical sets which results in the same set $$x$$. So $$x = x + 1$$ would be true here. But then sadly I guess sets don't quality as numbers? It fit so well that I wanted to include it though.

Oh, another one:

If you define $$=$$ as binding stronger than $$+$$ then technically, the equation $$x = x$$ is of course true for any number and the $$+1$$ can just be ignored

All of these approaches seem a bit cheet-y to me. I'd love to see a more elegant solution of this.

• thanks for the great answer. It would be accepted, but the question is not so good and I am likely to delete it. – tom Sep 29 '19 at 16:30
• @tom That's ok. Would you mind telling what solution you personally had in mind before doing that? – palsch Sep 29 '19 at 16:31
• sure no problem. For me the answer the computer scientist had in mind was infinity. I had not realized all the potential mathematical subtleties such as the answer in terms of set theory. – tom Sep 29 '19 at 16:40

Well...

No such number exists that satisfies $$x=x+1$$. If $$x=\infty$$ then it is possible however $$\infty$$ is not a number.

• "Keep in mind that infinity is not a number – Adam 25 mins ago" - Why did you post then? – Duck Sep 27 '19 at 23:48
• @Duck wording of - "Could anyone suggest a number for x?" – Adam Sep 27 '19 at 23:50
• "No such number exists that satisfies $x = x + 1$" Do you have a proof for that? Or is it just "obvious"? – palsch Sep 27 '19 at 23:52

This is probably a bit too 'on the computer programming side' to be the mathematicians response, but still...

What about $$x=\frac{1}{9}$$? If $$+$$ is seen as the concatenation operation, then $$x=0.\overline{1}=0.\overline{1}+1$$.

• Welcome to puzzling SE - sorry that the question is not so good and I am likely to delete. I think your answer is quite ingenious and would upvote it, but as I am going to delete the question there is no point in upvoting I'm afraid. - Well done for using the format for hiding your answer! I hope that you enjoy Puzzling SE and consider answering other better questions. – tom Sep 29 '19 at 16:33
• ok, see the comments by the question, but this question probably won't be deleted, so I upvote for the nice answer – tom Sep 29 '19 at 20:18

• I think the question aimed at not defining the $=$ like a programmer since a programmer asks a mathematician about his point of view, not the other way around. – palsch Sep 27 '19 at 23:55