There are two possible extensions of the rational numbers: the reals and the p-adic numbers. This is because of Ostrowski’s theorem which states that any norm d

There are two possible extensions of the rational numbers: the reals and the p-adic numbers. This is because of Ostrowski’s theorem which states that any norm defined for rationals, can either be the normal absolute value or the p-adic norm. For p being a prime, n and d being integers relatively prime to p and e being an integer, a rational can be written as r=p^e*n/d. And p-adic norm is p^(-e). Then in 3-adic numbers you can write 1/2 = 2 + 3 + 3^2 + 3^3 + 3^4 + … which actually converges because the norm has the negative of the exponent. pari/gp has a native p-adic calculator, where you can type 1/2 + O(3^5) and get the first p-adic digits. I don’t know why this impresses me, but it is probably because there are just two ways of extending the rationals, and the p-adic numbers are like the lesser known brothers of reals. I feel, I should have known this for ages. And no, I don’t know if there are known applications, outside number theory, but still, its cute.