Continued fractions

A simple continued fraction is an expression of the form $a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}}$ where $a_0$ is an integer and all $a_i$ for $i \geq 1$ are positive integers. Every rational number can be written as a finite continued fraction, and some irrational numbers have infinite continued fraction representations.

Important properties

• Continued fractions give the best possible rational approximations to real numbers.

• The process of converting a fraction to a continued fraction is related to the Euclidean algorithm.

• The convergents (the values you get by stopping after a certain number of terms) get closer and closer to the actual value.