Continued fractions are particular fractions: they are fractions whose denominator continues its descent.
where a are integer numbers. Continued fractions were studied yet in ‘500 by Bombelli and Cataldi. A systematical study began in ‘700 with Euler and it continued with Jacobi, Lagrange, Galois and Gauss.
Let us start to study some interesting properties. A finite continued fraction represents a rational number:
We can consider infinite continued fractions too. Infinite continued fractions represent irrational numbers. Given any number (rational or irrational), it’s possible to obtain its expansion in continued fraction through this algorithm:
The algorithm start for k=0 and it gives in output the a of the continued fraction expansion of a starting number called alpha_0. The symbol [*] indicates the floor of the number. If the starting number is irrational, the algorithm does not end.
The continued fraction representation of an irrational number is really amazing. Euler and Lagrange showed that a periodic infinite continued fraction represent ever a quadratic irrationality and viceversa. For example:
This fact is extraordinary. Usually we think to irrational numbers to something that has no rationality and regularity. For example the square root of 2 is the number 1,41421356237…
Instead with continued fractions we can write with regularity every quadratic irrationality. In 1848 Hermite posed this problem:
“Find methods for writing numbers that reflect special algebraic properties”
This problem is still unsolved.