Approximation by Rationals

choose x

1.41421

choose q

3

calculate p automatically

find p such that p/q ∈ (x-1/2q, x+1/2q)

1

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

x

and natural number

q

, there is a natural number

p

such that

|x-p/q|<1/2q

.

This can be seen by partitioning the real line into the half-open intervals

[p/q,(p-1)/q)

of length

1/q

,

p∈

. The number

x

must fall into such an interval.