# Does .99(repeating) = 1

**URL:** <https://boards.straightdope.com/t/does-99-repeating-1/170821>\
**Category:** Factual Questions\
**Created:** [April 24, 2003, 11:03am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821 "2003-04-24T11:03:48Z")\
**Posts on this page:** 2\
**Page:** 6

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**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\
**Post date:** [April 26, 2003, 4:25am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/101 "2003-04-26T04:25:50Z")

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> [@](#):
>
> \*Originally posted by Orbifold \*  
> \*\*I think twos-complement notation owes more to plain old modular arithmetic than to p-adic arithmetic. A 8-bit integer register (for example) is just performing arithmetic modulo 2[sup]8[/sup], which is why -1 is represented as 2[sup]8[/sup]-1, or 11111111 in binary. There is probably some way in which 2-adic arithmetic can be thought of as the “limit” of arithmetic modulo 2[sup]n[/sup] as n goes to infinity, and a number theorist could probably explain it detail, but I can’t.
> 
> It’s certainly an interesting correspondence, though. \*\*

Since the issue of the OP seems to have been settled amicably, I thought I would continue this interesting sidelight. It turns out that 2-adic arithmetic (and p-adic arithmetic for other primes p) is interesting to topologists as well as number theorists. And **Orbifold** is entirely correct that it is the limit of modular arithmetic (in modern terminology; in slightly older terminology it is the _inverse_ limit). One way to look at it is to consider infinite sequences (a[sub]1[/sub], a[sub]2[/sub], a[sub]3[/sub], …) where each a[sub]n[/sub] is an integer modulo 2[sup]n[/sup] and a[sub]n-1[/sub] is congruent to a[sub]n[/sub] modulo 2[sup]n-1[/sup]. (That is, it is the remainder on dividing a[sub]n[/sub] by 2[sup]n-1[/sup].) If you were to stop your sequences at the nth term, looking at only (a[sub]1[/sub], a[sub]2[/sub], …, a[sub]n[/sub]), then the whole sequence would be determined by the last term, a[sub]n[/sub], and you would be looking at just arithmetic modulo 2[sup]n[/sup]. However, when you “take the limit” and allow yourself to look at infinite sequences, you get a new beast, the 2-adic numbers.

Now write 1 for (1, 1, 1, …), 2 for (0, 2, 2, 2, …), 4 for (0, 0, 4, 4, 4, …) and so on. Then 1 + 2 + 4 + 8 + … = (1, 3, 7, 15, …) = (-1, -1, -1, …) = -1. 🙂

And let me repeat **Orbifold** ’s warning: This has nothing to do with infinite sums of real numbers. In fact, it’s almost the opposite, in some sense. For those mathematicians in the audience worried about the infinite sums above, there is a topology around (even a metric, if I remember correctly) in which those sums converge. In fact (again, IIRC) we’re looking at the completion of the integers with respect to this metric. But that point of view obscures the connection with modular arithmetic.

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**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\
**Post date:** [April 26, 2003, 10:13am UTC](https://boards.straightdope.com/t/does-99-repeating-1/170821/102 "2003-04-26T10:13:27Z")

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> [@](#):
>
> \*Originally posted by RM Mentock \*  
> \*\*[I always knew that there was some sort of ulterior motive](http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=22371&perpage=40&pagenumber=1)! 😉 Wow, those old threads get all balled up don’t they? \*\*

Thank you, Father. That might have been the thread I was looking for. The OP seems to be missing, though. It was specifically your excellent and compelling reasoning that caused me to change my mind.

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