Irrational number approximation by a rational number

Is there a way ( a theorem ) to find a rational number for a given irrational number such that it is an approximation to it to the required decimal places of accuracy. For example 22/7 is an approximate for pi for 2 decimal places.

Answers and Replies

best rational approximation (via continued fractions)

mathman
Science Advisor
There is always the trivial solution. Write out the decimal expansion for the irrational to the required number (plus one) of places. Then truncate (or round). This rational number will satisfy the criterion.

lurflurf
Homework Helper
^That tends to give big denominators, how distasteful.

mathman
Science Advisor
^That tends to give big denominators, how distasteful.

Most of the time you can't do much else.

Write down the decimal expansion to the accurate number of digits required+1 and make the penultimate digit to round figure and then convert it to fraction by dividing it by 10 to power number of decimal digits.