Irrational number approximation by a rational number

[n.karthick]

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.

 

[Xitami]

best rational approximation (via continued fractions)

 

[mathman]

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]

^That tends to give big denominators, how distasteful.

 

[mathman]

^That tends to give big denominators, how distasteful.

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

 

[Count Iblis]

Mathematica has the command "Rationalize" to do this. I think it uses continued fractions as Xitami mentioned above. If you have more than one irrational number then you can ask if there is an approximate relation between them involving rational numbers. That problem can be efficiently solved using the LLL algorithm:

http://en.wikipedia.org/wiki/Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm

 

[Abu Rehan]

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.