The irrationality of algebraic numbers like or is rather trivial, and so we can jump straight to the problem of irrationality measures in the sense of Chapter 8. One argument is fast, using the notation of the previous chapter. Namely if is near *r*/*s* (with *r* in Z and *s* in N), then *ω* = *r*−*sα* is small and as soon as |*ω*| ≤ 1 we get |*r*| ≤ (*α*+1)*s* so ||*ω*|| ≤ |*r*|+*s*||*α*|| ≤ (2*α*+1)*s*. Thus by (11.9) with *d* = 2 we get

Start Free Trial

No credit card required