> But how long will we need to look through these sequences of digits before we find the disagreeing digit? It feels intuitively like we should be able to establish some kind of bound on this. Like, maybe we should be able to say “if you add two lists of n numbers, each of which has d digits, then they can’t disagree for more than k * n * d digits” for some k. But no-one’s been able to prove anything like this.
You can write down a completely explicit bound here using the Mahler-Mignotte root separation bound. More generally, for any algebraic expression involving algebraic numbers, you can bound a priori the number of digits you need to check to determine its sign.
When you involve transcendental numbers, things do get much harder though.
You can write down a completely explicit bound here using the Mahler-Mignotte root separation bound. More generally, for any algebraic expression involving algebraic numbers, you can bound a priori the number of digits you need to check to determine its sign.
When you involve transcendental numbers, things do get much harder though.