The fraction is a good approximation to the irrational number : so much so that it is celebrated in Pi Approximation Day on 22 July. But have you ever wondered how to calculate rational approximations to irrational numbers?
The answer comes from continued fractions: these are a nested series of fractions that can reveal hidden properties of numbers. Any number can be written as a continued fraction. Rational numbers (including integers) can be written as finite continued fractions: for example
The first few approximations of produced in this way:
very quickly get close to the true value of . Their differences from are approximately:
|If you count the curves of seeds spiralling in a sunflower you'll find pairs (counting spirals curving left and curving right) that are (almost always) neighbours in the Fibonacci series, all approximations of .|
But not all approximations work so well for irrational numbers. For example, the approximations from the continued fraction of (the limit of the ratio of successive Fibonacci numbers – you can read more here) are:
You can read more about continued fractions in John D Barrow's article, Chaos in Numberland: The secret life of continued fractions.