Fractions produced by mediants have some very interesting properties. We saw some of them in connection with the Stern-Brocot tree. This articles explores a more curious property, relating fractions to circles in the plane. It was discovered in 1938 by Lester R. Ford and is also mentioned in Conway and Guy’s The Book of Numbers.
Archives for number theory
The Stern-Brocot Tree of Fractions
Consider two fractions \frac{m_1}{n_1} and \frac{m_2}{n_2} with positive numerators and denominators. The fraction \frac{m_1+m_2}{n_1+n_2} is called the mediant of \frac{m_1}{n_1} and \frac{m_2}{n_2}. It is straightforward to show that the mediant is placed numerically between the original fractions,
Continued Fractions and Continuants
We will be considering continued fractions of the form
a_0 + \displaystyle\frac{1}{a_1 + \displaystyle\frac{1}{\ddots + \displaystyle\frac{1}{a_{n-1} + \displaystyle\frac{1}{a_n}}}}
Computing the Greatest Common Divisor
The greatest common divisor of two integers is the largest positive integer that divides them both. This article considers two algorithms for computing \hbox{gcd}(u,v), the greatest common divisor of u and v.
