Explanation of the 5-adic expansion of 2

Question

Why is the $5$-adic expansion of $2 = 2 + 0\cdot5^2 + 0\cdot5^3 + ...$?

I've done some working with powers of $5^k$ for $k=1,2,...$ and got that the 5-adic expansion is $2+2\cdot5+2\cdot5^2 + 2\cdot5^3 + ...$.

An explanation would be appreciated!

Answer 1

It doesn’t seem that you understand the $p$-adic expansion. Running the formula for the sum of a convergent geometric series on your expansion, we have $a=2$, $r=5$, so your number is $2/(-4)=-1/2$.

The expansion of any rational number $\lambda$ must be $\sum_{i\ge0}a_ip^i$, with each $a_i$ in the range $\{0,\cdots,p-1\}$, in such a way that for each $n$, $\lambda\equiv\sum_0^{n-1}a_ip^i\pmod{p^n}$. In your case, you need $a_1$ so that $2+a_1\cdot5\equiv2\pmod{25}$. I hope that you see that the correct value of $a_1$ is zero, but that your value of $2$ would claim that $2+2\cdot5=12\equiv2\pmod{25}$. But twelve and two are not congruent modulo $25$.

Answer 2

Your proposed answer $$ 2+2\cdot5+2\cdot5^2 + 2\cdot5^3 + ... $$ is a geometric series; it converges in the $5$-adic metric to: $$ \frac{2}{1-5} = -\frac{1}{2} $$ so its value is not $2$, as desired.