Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Plus Advent Calendar Door #10: Continued fractions

      10 December, 2016

      The fraction 22/7 is a good approximation to the irrational number π: 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 3=3 (which is admittedly not very interesting!) 227=3+17 and 333106=3+17+115=3+15106. Irrational numbers have infinite continued fractions, and cutting these off after a finite number of levels gives you fractions that are close to the value of the irrational number. The fractions above are the first few such approximations to π, calculated from cutting off its infinite continued fraction: π=3+17+115+11+1292+…. A fraction p/q is a good approximation of an irrational number x if there isn't a fraction that is closer to x with a smaller denominator q. And the continued fraction expansion of an irrational number gives the best series of approximations in this sense, for an irrational number.

      The first few approximations of π produced in this way: 3,22/7,333/106,355/113 and 103993/33102 very quickly get close to the true value of π. Their differences from π are approximately: 0.141,0.001,0.0008,0.0000003 and 0.0000000006.

      Sunflower seed head Sunflower seed head with 55 left spirals Sunflower seed head with 34 right spirals
      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 ϕ.

      But not all approximations work so well for irrational numbers. For example, the approximations from the continued fraction of ϕ=1.618... (the limit of the ratio of successive Fibonacci numbers – you can read more here) are: 11,21,32,53,85,… These approximations, although the best there are, approach the true value for ϕ very slowly, with differences of around 0.618,0.381,0.118,0.048 and 0.018 for those listed above. Irrational numbers with such useless approximations are called badly approximable. You can tell something is badly approximable if the numbers in its continued fraction are bounded (they don't ever exceed some fixed number) and this is a measure of how irrational they are. In fact, ϕ is the most irrational number, with the numbers in its continued fraction never exceeding 1: ϕ=1+11+11+11+11+11+11+…. The continued fraction for ϕ reveals a beautiful pattern, despite the fact that it (and all irrational numbers) having messy infinite decimal expansions that do not tail off into any repeating patterns. The continued fractions of other irrational numbers, such as 2, also reveal hidden patterns: 2=1+12+12+12+…. These patterns are beautiful, but result in these numbers being badly approximable. But π, with its excellent rational approximations, holds no such pattern in its continued fraction.

      You can read more about continued fractions in John D Barrow's article, Chaos in Numberland: The secret life of continued fractions.

      Return to the Plus advent calendar 2016.

      This article comes from our Maths in a minute library.

      • Log in or register to post comments

      Read more about...

      advent calendar 2016
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms