As an example to illustrate our point. Take a look at the well known geometric
S(N) = Σk=0N rk
= 1 + r + r2 + r3 + r4 + r5
+ r6 + r7 ... = (1 - rN)/(1 - r)
Suppose r = 1/2 and use binary notation in the sequel.
Then S(0) = 1 , S(1) = 1.1 , S(10) = 1.11 , S(11) = 1.111 , S(100) = 1.1111
and so on.
The exact outcome is 10 (binary). Thus we may conclude that:
10 = lim N → ∞
1.1111111111111111111111111111111111111111111111111 (N bits) ...
The deviation from the exact outcome can be made as small as we want.
For N = 15 (decimal) = 1111 (binary):
10 - 1.111111111111111 ≤ 0.000000000000001
The point is that our exact value can only be reached if N approaches infinity.
Thus an unlimited accuracy with the rational numbers is accompanied inevitably
with an unlimited magnitude of the natural numbers.
And vice versa.