So, there I was, happily working out that the square root of 5,100 is very close to 70+2

OK, OK, OK, big aside here. Why is that? I’ll tell you why that is. It’s due to the binomial expansion, in a slightly surprising way:

  • (5000+x)1/2502+x1002
  • (5000x)1/2502x1002
  • So 510049001001002+1001002
  • Or 5100702.

Lovely stuff. But then in stepped in @shalock, who noted that it’s not even the best such approximation. That would be 132635+2.

Now that – that just seems wrong. We can do something sort of similar, based around 1275.125 – the square of 10124:

  • (1275.125+x)1/210124+x2101
  • (1275.125y)1/210124y2101
  • So 132612254072808+4012808
  • Which is, again, approximately 2.

I haven’t worked through the higher-order terms, on the grounds that they’re awful. The x2 terms in the expansions around 5000 cancel out; the x2 terms around 1275.125 don’t cancel (because the xs are different), but I suspect they might balance out with the x3 terms to make a closer approximation.

As for why 1326 and 5100 are very close to 35 apart, I don’t know. Any ideas?