# Tag Info

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### Why does an argument similiar to 0.999...=1 show 999...=-1?

If you want to understand the mathematics behind these things, it is all based upon the notions of 'convergence' and of 'limits'. If you read any first course in analysis textbook you will find the ...
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### Why does an argument similiar to 0.999...=1 show 999...=-1?

In the $10$-adic numbers it is true that $\dots 9999 = -1$. More precisely, the series $\sum_{n=0}^{\infty} 9 \cdot 10^n$ converges in $\mathbb{Q}_{10}$ and its limit there is $-1$.
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### Why does an argument similiar to 0.999...=1 show 999...=-1?

As other users have noted, it doesn't make sense to have infinitely many nines to the left of the decimal point. This is because the sequence $9, 99, 999, \ldots$ doesn't converge to anything, unlike ...
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This must be covered in almost every text on the $p$-adic numbers; I think the book of Gouvêa is the best of these. And your statement is not quite true: for $p=2$, $\mu_{p-1}$ is trivial all right, ...
• 60.5k
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### Ring of the integer $p$-adic numbers $\mathbb{Z}_p$

Let's look at a different ring first: $R=k[[X]]$, the ring of formal power series over a field. We can also think of $R$ as the ring of sequences $f_0, f_1, f_2,\ldots$, where each $f_i$ is a ...
• 29.4k
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### Method of finding a p-adic expansion to a rational number

The short answer is, long division. Say you want to find the $5$-adic expansion of $1/17$. You start by writing $$\frac{1}{17}=k+5q$$ with $k \in \{0,1,2,3,4\}$ and $q$ a $5$-adic integer (that is, ...
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### Why does an argument similiar to 0.999...=1 show 999...=-1?

One thing to think about is how this plays with modular arithmetic, if you're familiar. Basically, arithmetic mod $10^n$ is arithmetic where we only care about the last $n$ digits of a number and is ...
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