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
    • Maths in a minute: the Fibonacci sequence

      1 May, 2020
      3 comments
      Leonardo Fibonacci c1175-1250.

      Leonardo Fibonacci c1175-1250.

      The Fibonacci sequence

      1,1,2,3,5,8,13,21,34,55,89,... is one of the most famous number sequences of them all. We've given you the first few numbers here, but what's the next one in line? It turns out that the answer is simple. Every number in the Fibonacci sequence (starting from 2) is the sum of the two numbers preceding it: 2=1+13=1+25=2+38=3+5, and so on. So it's pretty easy to figure out that the next number in the sequence above is 55+89=144, and (in theory at least) to work out all numbers that follow from here to infinity.

      Hear the music of the Fibonacci sequence in our podcast!

      Where does it come from?

      The Fibonacci is named after the mathematician Leonardo Fibonacci who stumbled across it in the 12th century while contemplating a curious problem. Fibonacci started with a pair of fictional and slightly unbelievable baby rabbits, a baby boy rabbit and a baby girl rabbit.

      rabbits

      They were fully grown after one month

      rabbits

      and did what rabbits do best, so that the next month two more baby rabbits (again a boy and a girl) were born.

      rabbits

      The next month these babies were fully grown and the first pair had two more baby rabbits (again, handily a boy and a girl).

      rabbits

      Ignoring problems of in-breeding, the next month the two adult pairs each have a pair of baby rabbits and the babies from last month mature.

      rabbits
      Fibonacci asked how many rabbits a single pair can produce after a year with this highly unbelievable breeding process (rabbits never die, every month each adult pair produces a mixed pair of baby rabbits who mature the next month). He realised that the number of adult pairs in a given month is the total number of rabbits (both adults and babies) in the previous month. Writing An for the number of adult pairs in the nth month and Rn for the total number of pairs in the nth month, this gives An=Rn−1. Fibonacci also realised that the number of baby pairs in a given month is the number of adult pairs in the previous month. Writing Bn for the number of baby pairs in the nth month, this gives Bn=An−1=Rn−2. Therefore, the total number of pairs of rabbits (adult+baby) in a particular month is the sum of the total pairs of rabbits in the previous two months: Rn=An+Bn=Rn−1+Rn−2. Starting with one pair, the sequence we generate is exactly the sequence at the start of this article. And from that we can see that after twelve months there will be 144 pairs of rabbits.

      Where does it go?

      Real rabbits don't breed as Fibonacci hypothesised, but his sequence still appears frequently in nature, as it seems to capture some aspect of growth. You can find it, for example, in the turns of natural spirals, in plants, and in the family tree of bees. The sequence is also closely related to a famous number called the golden ratio. To find out more visit our collection of articles about Fibonacci and his mathematics. And if you'd like to hear how the Fibonacci sequence sounds you should listen to our podcast!


      About the author

      Rachel Thomas is Editor of Plus.

      • Log in or register to post comments

      Comments

      jennifer yusi

      17 July 2020

      Permalink

      The rabit as a cover of this page, which attracts me to open the post. The number of rabbit is a good way to apply in mathmatics. That's funny!

      • Log in or register to post comments

      Steve

      31 July 2020

      Permalink

      If you start with 0 and 1 then every number after those is the sum of the two numbers preceding it.

      • Log in or register to post comments

      Shariputra

      22 September 2020

      Permalink

      The Fibonacci sequence was actually discovered much earlier by Indian poets and linguists who came across this sequence while studying Sanskrit poetry. More information on this can be found, for example, on the wikipedia page for the Fibonacci numbers: https://en.wikipedia.org/wiki/Fibonacci_number. This topic is also discussed by the Fields Medallist Manjul Bhargava in multiple lectures on YouTube.

      Hence, in this article, under the heading 'Where does it come from'?, it might be pertinent to mention that this sequence was in fact discovered much earlier. This might help dispel the misunderstanding that Fibonacci was the first one to come across this sequence. It will also enrich the article by showing the readers how this sequence arose from a completely different (and surprising!) context- poetry!

      • Log in or register to post comments

      Read more about...

      Maths in a minute
      Fibonacci
      Fibonacci number
      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