Approximation by Rationals
Approximation by Rationals
This Demonstration illustrates that any real number can be approximated by a rational number to within an error equal to half its denominator. In symbols, for any real and natural number , there is a natural number such that .
x
q
p
|x-p/q|<1/2q
This can be seen by partitioning the real line into the half-open intervals of length , . The number must fall into such an interval.
[p/q,(p-1)/q)
1/q
p∈
x