The ancient Greeks wondered when irrational numbers, like pi, can be represented with fractions. Two mathematicians now have a complete answer.

First, decide on how close the approximation should be for fractions of a particular denominator. (Remember, the “numerator” refers to the top of a fraction and the “denominator” the bottom.

This is the business of rational approximation. Ancient mathematicians, for instance, recognized that the elusive ratio of a circle’s circumference to its diameter can be well approximated by the ...

Scientists have calculated pi to 105 trillion digits, although most of us are more familiar with the approximation 3.14. But how do we know that pi is an irrational number?

They realized, however, that there were approximations, e.g., the fractions 25/8, 22/7, 256/81, etc., that were close, and these fractions were employed for centuries as substitute for pi.

Kurt Spielberg, Efficient Continued Fraction Approximations to Elementary Functions, Mathematics of Computation, Vol. 15, No. 76 (Oct., 1961), pp. 409-417 ...

S. ITO and H. NAKADA, On natural extensions of transformations related to Diophantine approximations, Proceedings of the Conference on Number Theory and Combinatorics, Japan 1984, World Scientific ...