Fibonacci
The amazing number sequence
Leonardo Bonacci (c1170 - c1250) - also known as Fibonacci - was an Italian mathematician.
His name is forever attached to a fascinating sequence of numbers - the Fibonacci Sequence - 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610… and so on forever. After the initial 0 and 1, each number in the sequence is the sum of the two numbers immediately preceding.
Please watch this short video -
Fibonacci and the Golden Ratio:
The Fibonacci sequence is closely connected to the golden ratio - (earlier Viewpoint). If we work out the ratio of successive numbers in the Fibonacci sequence we get ever closer to the golden ratio - 1.6180339887498948482... etc.
For instance, 377/233 = 1.6180258 and 610/377 = 1.6180371.
The first 300 Fibonacci numbers are shown HERE. If we take the 30th number in the list (= 832040) and divide it by the 29th in the list (= 514,229), the answer is 1.618034.
Dr Ron Knott (formerly of Surrey University) has put together an online resource aimed, in his words, at ‘those who don't like Mathematics or who hated maths at school as well as for teachers and those at school (or who left school a long while ago!) who want to see a fun side of maths and who like to play with numbers.’
Why not delve into Dr Knott’s webpages which do not require more than basic mathematical knowledge - Ron Knott's Mathematics Pages and Contact details (surrey.ac.uk)
Fibonacci numbers are covered by Dr Knott’s webpages - Fibonacci Numbers, the Golden section and the Golden String (surrey.ac.uk) and The Mathematical Magic of the Fibonacci Numbers (surrey.ac.uk).
Pythagorean Triples and Fibonacci:
The Pythagorean theorem (previous Viewpoint) tells us that a right-angled triangle is formed from sides 3, 4 and 5 units. Another combination (“triple”) is 5, 12, 13.
These are known as Pythagorean Triples and they are connected to Fibonacci numbers. Various examples are shown in this Table
Finding a Pythagorean Triple from four consecutive Fibonacci numbers:
Call the four consecutive numbers D, E, F, G
The first number in a triple is D x G.
The second number in the triple is 2 x (E x F)
The third number in the triple is E2 + F2
e.g. Take the four consecutive Fibonacci numbers 2, 3, 5, 8
The first number in a triple is 2 x 8 = 16
The second number is 2 x (3 x 5) = 30
The third number is 32 + 52 = 34
Hence, 16, 30, 34 is a Pythagorean Triple.
Use a calculator to show that 162 + 302 = 342
(Multiplied out it is 256 + 900 = 1156 and 342 = 1156).
Further on Fibonacci and Pythagorean Triples:
A video (from the Mathologer series on Youtube) looks at the connection between Pythagorean Triples and Fibonacci. Although it is 42 minutes in length it is a good insight into an interesting pattern and well-worth watching.
Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion - YouTube
(PDF) Fibonacci Meets Pythagoras (researchgate.net)
Check whether a given number is in the Fibonacci sequence:
It can be proved that a number N is a Fibonacci number if and only if either
5N2 + 4 or 5N2 – 4 is a perfect square!
Is 21 a Fibonacci Number? It is but let’s test it -
5 x 212 + 4 = 2209 or 5 x 212 - 4 = 2201.
The square root of 2209 is 47 and so 2209 is a perfect square (47 x 47 = 2209).
21 is therefore a Fibonacci number. (The square root of 2201 is approximately 46.915).
Music:
It would not be right to leave the Fibonacci sequence without mention of music and how composers and musical instrument makers have used the sequence and the Golden Ratio to compose and create music -
What is the Fibonacci Sequence – and why is it the secret to musical greatness? - Classic FM.
Links:
Special sequences - Sequences - AQA - GCSE Maths Revision - AQA - BBC Bitesize
What is the Fibonacci sequence? | Live Science
Fibonacci Sequence - Definition, List, Formulas and Examples (byjus.com)
Fibonacci Numbers, the Golden section and the Golden String (surrey.ac.uk)
Pythagorean Triples ( Definition, Formula, List, and Examples) (byjus.com)
Extra February 2024:
Corridor numbers, Fibonacci numbers, and Pascal’s triangle (substack.com)
