Stream: Miscellaneous

Topic: An infinite sum

Julin S (Jun 27 2022 at 07:22):

I recently came across this infinite summation:

1 + 2 + 4 + 8 + 16 + 32 + 64 + ...

And how a naïve way like

         S =  1 + 2 + 4 + ...
=>      2S =  2 + 4 + ...

=>  2S - S = -1
=>       S = -1

can make it look as if the sum ends up being -1.

I heard someone saying that this would make sense in the p-adic number system with p=2.

Does anyone know how that would be?

Rodolphe Lepigre (Jun 27 2022 at 07:54):

You might find this excellent video interesting: https://youtu.be/XFDM1ip5HdU.

Li-yao (Jun 27 2022 at 09:13):

p-adic numbers can be represented as decimal expansions that are infinite "to the left", so there is no most significant digit. In that system, if you consider the number x = ....1111, whose digits are all 1s, then adding 1 gives you x+1 = ....0000 = 0 (you keep carrying so all the digits are 0) therefore ...1111 = -1.

Julin S (Jun 29 2022 at 10:05):

Could this be a possible direction?

Don't know much of p-adic, except for this in the case of integers:

𝑉ₚ(x) = largest power of p that divides x

(not sure what this function is called.)

ie, greatest mfor which

n ≡ 0 mod pᵐ

Absolute value calculated as:

|x|ₚ = 1 / p^(𝑉ₚ(x))

So the series

1 +  2  +  4  +  8  +  16  +  32  + ...

in 2-adic number system would be

     1     1     1     1      1
1 + --- + --- + --- + ---- + ---- + ...-
     2     4     8     16     32

Now we got a converging series.

But how would we go from here?

Would addition be still as usual in 2-adic?

Jason Rute (Jun 30 2022 at 02:55):

This conversation is going on in two Zulips, but this one is has more people so I’ll continue here.

Jason Rute (Jun 30 2022 at 02:55):

In general 2-adic numbers are an extension of the integers just like the reals are an extension of the integers. But they are not the reals. As Li-Yao said one way to think of them is as follows. Every nonnegative integer can be represented as a binary number, e.g. 101011. If we allow finitely many 0s and 1s to the right of that number (and to the right of the decimal place) we get the nonnegative dyadic rationals, e.g. 101011.0111. If we allow infinitely many 0s and 1s then we get the nonnegative real numbers, e.g. 101011.011101… Now, the 2-adic numbers (p=2 since we are working in binary) are the same idea but we start with the binary integers and add infinitely many 0s and 1s to the left of the number, e.g. …10101010. (See li-Yao’s answer for more details.)

Jason Rute (Jun 30 2022 at 02:55):

I don’t think your math above is correct. Adding x + y in the 2-adics isn’t the same as adding |x|_p + |y|_p which I think is what you are doing.

Jason Rute (Jun 30 2022 at 03:06):

@Julin S Indeed the point of the p-adic numbers is that they aren’t the usual real or complex numbers, so you can’t just translate them to that world. Instead they are a new set of numbers all together. I asked in the other Zulip if you knew abstract algebra since that concerns the idea of abstracting away the important properties of numbers to get at lots of other number-like systems which still have the usual operations like addition. (Topology is another similar subject, but it concerns the idea of abstracting away the notion of space where you still have the important notions of closeness and limit. The p-adic numbers mix ideas from both algebra and topology). You might be interested in this video on abstract algebra which helps to appreciate this abstraction. https://m.youtube.com/watch?v=mH0oCDa74tE

Last updated: Oct 01 2023 at 19:01 UTC