# 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