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Published 1999 Revised 2016
Continued fractions are written as fractions within fractions which are added up in a special way, and which may go on for ever. Every number can be written as a continued fraction and the finite continued fractions are sometimes used to give approximations to numbers like \sqrt 2 and \pi .
To see how to work out a continued fraction let X = {1\over\displaystyle 2\;+\; {\strut 3\over \displaystyle 4 }}.
Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?