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Working with p-adic integers is really simple, and usually if you want to do rational numbers, it's fairly easy; you just use long-division.

But here in the case 9/10 when we're working in the 5-adics; I have no idea. I understand you can extend the p-adic integers to the p-adic numbers; but given $9 = 4 + 1 \cdot 5 + 0 \cdot 5^2 + ...$ and $10 = 0 + 2 \cdot 5 + 0 \cdot 5^2 + ...$ we can't do long-division. So it seems that we'd have to get it in p-adic number form; the issue here is though that I have no idea how to do this. Is it simply trial in error? If so, the question after this is $\frac{1234}{625}$ how the hell am I meant to do that!!!!!

J. W. Tanner
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KeynesianSpaceman
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  • In the set of 5-adics numbers, all primes different from 5 are invertible, while 5 is not invertible. In particular dividing by 5 is easy: it's simply shifting the point. For example $$\frac{2}{5}= 0.2$$ $$\frac{4}{5^3}= 0.004$$ – Crostul May 16 '23 at 14:29
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    Paradoxically 1234/625 is easy since $625=5^4$ and $1234 = 14414_5$ so the answer is $1.4414_5$ – Crostul May 16 '23 at 14:35
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    Cf. https://math.stackexchange.com/q/1186967/96384 and many similar questions on this site. – Torsten Schoeneberg May 17 '23 at 03:08
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    In particular https://math.stackexchange.com/q/3600639/96384. – Torsten Schoeneberg May 17 '23 at 13:51

1 Answers1

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Note that $1/5$ (or any power of it) is not a $5$-adic integer. It is just $\ldots0000.1$, a $5$-adic rational.

Since the geometric series gives $\ldots1111=\sum_n5^n=1/(1-5)=-1/4$, we have

$$\ldots2222=-1/2$$ $$\ldots2223=(-1/2)+1=1/2$$

so

$$9/10=(9/2)/5=(4+1/2)/5=(\ldots2232)/5=\ldots2223.2$$

For the second question, the key is that $625=5^4$ is a power of $5$. So

$$1234/625=0.1234$$

In general, you can simply shift the radix point for each factor of $5$ in the denominator; and the remaining factors in the denominator must be factors of some $(5^m-1)$, so you can move them to the numerator by using the geometric series $\sum_n5^{mn}=1/(1-5^m)$.

mr_e_man
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  • Oops, I forgot that $1234$ is in decimal. @Crostul is correct. – mr_e_man May 16 '23 at 14:40
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    I think I found an easier way to calculate it, here I just noted that 9/10 = 2/5 + 1/2 and wrote 1/2 in 5-adic form, and I did similarly for 1234/625 by breaking it down, you're right though noting 625 = 5^4 really helped. – KeynesianSpaceman May 16 '23 at 15:02
  • Another way is $\frac{9}{10}+\frac{1}{10}=1$ so $\frac{9}{10}=1-\frac{1}{2}*\frac{1}{5}$. Since $-\frac{1}{2}=...222$ like above, you just need to shift the decimal place and add $1$ to get $...223.2$ – Merosity May 29 '23 at 23:53