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I don't think I understand how p-adic numbers relate to the usual concept of infinity. The wiki page and various sources on the internet did not help.

Let's see for example the 10-adic counterparts to the real numbers (10 is not prime, but it is more convenient here for demonstration purposes):

$$ 0 \rightarrow 0 \\ 1 \rightarrow 1 \\ 2 \rightarrow 2 \\ $$

Anyway, for any positive whole number we have the usual representation. On the other hand:

$$ -1 \rightarrow ...9999 \\ -2 \rightarrow ...9998 \\ -3 \rightarrow ...9997 \\ $$

But if $...9999$ is not equivalent to $+ \infty$ then where is the infinity? Is zero the infinity?

$$ -1+1=0 \rightarrow ...0000=0 \\ $$

Do negative and positive infinities 'meet in the middle' between $0$ and $-1$?

It also seems that many infinite series divergent in the field of reals, converge in the field of p-adic numbers. This question is about the divergent series, but I am more interested in the general concept of infinity for p-adics.

For now it seems to me that p-adics incorporate everything in the same form - whole numbers, rational and irrational numbers, negative numbers and even infinity.

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Yuriy S
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    Infinity, as commonly used in limits, can't be a p-adic number because the p-adic numbers $\mathbb{Q}_p$ form a field, and field axioms are incompatible with the property $\infty + a = \infty$. But as with any other field, you can define its projective line, $\mathbb{Q}_p \mathbb{P}^1 = \mathbb{Q}_p \cup {\infty}$. – pregunton Sep 13 '15 at 12:07
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    "I don't think I understand how p-adic numbers relate to the usual concept of infinity". I don't think I understand what you mean by "the usual concept of infinity". Consequently, this question makes no sense to me. – Torsten Schoeneberg Aug 10 '22 at 01:22

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