These pages on Mathematics are for 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. The level of mathematics required is
what would be taught up to GCSE (age 15) in UK schools and rarely does it need A-level skills (age 17, pre university).
Other pages are designed to help with A-level Maths topics, either as a teacher or student.
The Calculators on all pages here use only standard JavaScript, which is built-in to every browser. They should therefore
work on all browsers on all devices, mobile, tablet or PC.
The Fibonacci Numbers
Fibonacci Numbers
a collection of information on Fibonacci numbers (0,1,1,2,3,5,8,13,21,...)
and the Golden section ( 0.61803... and 1.61803...)
This Fibonacci page was originally developed in 1995 and went live on the web in March 1996 making it now one of the
longest running active pages on Mathematics on the web!
At the top it includes links to the pages shown here below. The whole site is hosted by the University of Surrey.
This page alone still gets more than 3000-5000 visits per day.
A simple way to reference this page is to use the automatic-redirection page
www.tinyurl.com/fibandphi.
Ron Knott was on Melvyn Bragg's In Our Time programme on BBC Radio 4, November 29, 2007
when we discussed The Fibonacci Numbers (45 minutes). You can listen again online or download the podcast.
Dr E Lawrence and myself, members of Surrey University Mathematics Department, were part of a large UK government
Teaching and Learning Technology Project (TLTP)
called Mathwise,
although now that name has been used by several other individuals and groups unassociated with this project.
It aimed to provide maths software to link school and university mathematics, involving
23 UK universities
developing activities and assessment on 40 topics in mathematics, ending in about 1995.
Mathwise never really reached its full potential because it was developed on both PCs and Macs using
a variety of software.
The PC and Mac software was
never integrated into one unified and accessible system.
I developed this Fibonacci page after Mathwise was ending, in 1995, to see if the (then new) internet
could be used to communicate maths effectively and generally, having the advantages that
it ran identically on many kinds of computer, both personal and desktop, PC and Mac (later: mobile and tablet)
it was accessible everywhere in any browser
it was free
it would not need extra software to be downloaded (later: nor extra apps)
it would not have any adverts except to buy the books referenced
later: it would not use cookies or any other way of keeping any data about readers
It was designed as both a resource for teachers as well as for keen and not-so-keen school and beginning-university students
who wanted to explore topics off the main curriculum. Hence each webpage is longer than is now become the norm
on the web so that
each could be downloaded and viewed at leisure offline, dating from the days
when a broadband connection was still rare, costly and very slow.
The pages gained many awards in its early days
and have been extensively expanded and augmented regularly since then.
Number bases: what happens if instead of using 10 as the basis of writing numbers we used base 2 (binary) or base −10?
What if we didn't use power of a number but used the Fibonacci numbers or the Factorials?
What about base Phi - the golden section number? or even a complex number?
Which fractions recur such as 1/3 =0.3333... and 1/7 = 0.142857 142857 ...?
HOw can we tell? How long is the recurring pattern for a given fractions?
Have you noticed that
1/99 = 0. 01 01 01 01 01 01 01 01 01 1 ... all ones - powers of 1 if you like
1/98 = 0. 01 02 04 08 16 32 65 30 61 2 ... the powers of 2 - with 'carry' when longer than 2 digits
1/97 = 0. 01 03 09 27 83 50 51 54 63 9 ... the powers of 3 - again with 'carry' if longer than 2 digits
1/9801 = 0. 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14.. the numbers in order!
Did you know that there is another recurring fraction for the powers of 2 but in reverse order...
1/199 = 0.00502512562814070351...145728 64 32 16 08 04 02 01 with a period of 99 digits.
Why is this? What other patterns can we find relating sequences and decimal fractions and why does this happen?
An accurate calculator to convert ordinary fractions (such as 2/7) to and from decimal fractions
(such as 0.285714...).
to any number of decimal places accuracy.
The web page does not need any extra software, just your browser on any device.
There are some lovely patterns of we take all the fractions made with the numerators and denominators 1 to n and put them in
order of size (Farey sequences). It leads us to a tree structure (the Stern-Brocot Tree) and answers some wuestions about
tightly packing circles on a flat surface.
explains how the
Egyptians and Babylonians of 3000 BC represented fractions and how they used them.
In some ways, their method is better than the decimal system! There are now some online calculators on this page to take some
of the work out of generating these fractions.
links their use in explaining the patterns on seedheads and flowers and their usefulness in mathematics too.
There is an online Continued Fraction Calculator
so you can experiment for yourself. No download is needed -- all you need is on the web page and in your browser (any browser)!
those integer-sided, right-angled triangles
such as the triangle with sides of 3, 4 and 5. Includes a formula for generating them all and calculator which
shows you how. These triangles were extensively studied by the Babylonians of
5000 years ago and some of the oldest mathematical writings (clay tablets) contain tables of such triangles.
The page includes several online interactive Calculators so you can experiment for yourself.
cos(60°)=0.5, sin(60°)=√3/2, tan(45°)=1. What other angles have simple expressions for their trig values?
cos(π/5) = Phi/2 where Phi is the golden section number (1+√5)/2
and lots of other facts and ways to remember the trig formulas.
Other ways of representing point with respect to a given triangle. Also a way to identify a given triangle point using
the Encyclopedia of Triangle Centers (ETC).
are sums of consecutive numbers, e.g. 4+5 is a runsum for 9, as is 2+3+4.
An on-line calculator computes all the runsums for a given number and finds numbers
with a specific number of runsums (e.g. under "2" would be 9 because 9 has just 2 runsums
shown above). Runsums are the difference between two Triangle Numbers, and this
is also explained on the web page.
We all know about square numbers and cubes but what about other shapes such as triangular, pentagonal (5 sided) or
tetrahedrons (a triangle-based pyramid) and square-based or other pyramids? More on Polygonal Numbers
Matchstick numbers, central polygonal numbers and some 3D and higher dimensional shapes.
A page on the amazing and fun maths facts, factoids and pictures on the number
2016, given at a talk with IMA NW Branch and MathsJam in Manchester in January 2016.
UK School MATHEMATICS RESOURCES for GCSE and A-Level
that generates variations on 50 questions, marks, detects common errors, and shows the correct
answer and provides bar charts of your results. It was originally developed for
the Mathematics Enhancement Programme at Exeter University as a resource for Mathematics teachers.
that can produce random cards, coins, dice, integers and items
for games or statistical experiments.
BigNumber JavaScript LIBrary (August 2019)
To accompany Michael Mclaughlin's excellent arbitrary-precision numbers
BigNumber.js for JavaScript, I have developed
a LIBrary of math functions including
integer functions (GCD, LCM, Factorial and Binomials and Random integers),
powers and Logs,
Trig functions and their inverses, Hyperbolic Trig functions and their inverses).
If you like the TV programme Countdown, you'll love Got It!. Select some number
cards and it will generate a target number for you to make using the cards and +, -, × and ÷ in
30 seconds. You can select the level of difficulty from primary school level to Maths MasterMind. IF you get stuck or run out
of time, Got It will show you one way to get the target.
which can not only find the day of the week for a given day, month and year, but
also tell you the years when your birthday falls on a Saturday, or which months in
a year have a "Friday 13th". It uses a simple table with no need for a calculator!
"Where do you go to get a degree in Apologies?" at the University
of Sorry (Surrey) (groan). Here's a collection of similar "courses".
About Dr Ron Knott
Ph.D(1980, University of Nottingham), M.Sc (1976, University of Nottingham), B.Sc (Pure Maths, University of Wales),
C.Math, FIMA, C.Eng, MBCS, CITP
Visiting Fellow, Department of Mathematics,
formerly Lecturer in Mathematics and Computing Science Departments (1979-1998)
Faculty of Electronics and Physical Sciences, University of Surrey,
Contact me initially by Email:
I was a lecturer in the Departments of Mathematics and Computing Science
at the University of Surrey, Guildford, UK, for 19 years
until September 1998 when I left to start working for
myself making web pages for maths education sites.
I now give mathematics talks to students at schools and universities
as well as to general audiences, teachers' conferences and Science Festivals
on topics of the web pages above:
especially the Fibonacci Numbers and why they occur so often in plants,
Fun with Fractions, As Easy As Pi, ... .
I now live in Bolton, near Manchester in NW England.
Upcoming and Recent talks, articles and events
2019 24 July, 14 September, 13 October University of Surrey Open Days
I will be hosting the "You do the Maths..." informal sessions throughout the day
in the Mathematics Department (top floor, AA Building)
Come and say Hi, try some hands-on maths and meet current students and staff.
It provides an automatic lookup of a centre using Peter Moses lists for the 6-9-13 triangle
and is self-contained with an introduction for those new to trilinear and barycentric coordinates.
2018 22, 23 June
2018 15 September The University of Surrey Open Days
"Is everything in Maths Always TRUE or FALSE? or When maths goes wrong!"
about puzzles and paradoxes in Logic suitable for a general audience but of
particular relevance to students applying to Maths Degree courses
These are free 15-20 minute "fun" talks on maths suitable for a general audience and held in the Mathematics Department
on top floor of the AA building: see
this PDF map of the campus.
On each day Ron will be giving three different presentations at 11:45, 12:30 and 14:15.
For more details of how to get to the University of Surrey in Guildford, here is some
travel information for train, coach and car.
Everyone welcome!
2017 6 November
Proceedings of the 2016 International Fibonacci Conference at Caen, France.
The 2016 Conference Proceedings
and all papers,
conference report, list of attendees and Problems Section
are now available online.
The 2018 Conference:
will be in Halifax, Nova Scotia, Canada, July 2-6, 2018. More details to follow soon.
2017 15 August
To celebrate one of the Pythagorean dates this year (15/8/17 in UK or 8/15/17 in USA)
the Pythagorean Triangles page has been updated with
a Sums of Squares Calculator
to find Pythagorean Quadruples and longer lists of square numbers whose sum is a square plus lots of
new Puzzles and Problem in a new section
now allow for very long integers.
2016 November 23:Talk at the University of Manchester Galois Group (student maths society)
2016 September 9 University of Surrey Open Day in Guildford: 3 free 15 minute
"mini-maths" fun talks in the Mathematics Department (AA Bulding, next to Sente House, top floor)
How to share Pizzas and Inheritances - two puzzles and what the ancient Egyptians can teach us today; A Date with Mathematics We live in an interesting year - mathematically! Some of the amazing maths facts
associated with the number 2016. Numbers you can Eat: the Fibonacci numbers and the golden ratio or putting the Phi into 'Fi-ve a day' and Fi-bonacci.
2016 June 24 and 25 University of Surrey (Guildford,UK) Open Days, Mathematics Department
Open to the general public, and three free 15 minute mini-maths talks for those who don't think Maths can be Fun!
2016 27 June - 2 July
a free presentation for the general public as part of
The International Conference on Fibonacci Numbers and their Applications
University of Caen, France
2016 February 6, Lancaster University Mathematics Masterclass, starts at 10am for those registered
2015 October 17 A repeat of the three talks given at the September University of Surrey Open Day,
held in
the Mathematics Department on Level 04 (the top floor) of the AA Building:
2015 September 12 The University of Surrey Open Day: Three 15 minute mini-maths talks for the general public
on How to share Pizzas and Inheritances - two puzzles and what the ancient Egyptians can teach us today; Dots and Flags or Proof By Pictures without any of that awkward algebra Numbers you can Eat: the Fibonacci numbers and the golden ratio
2015 March 21 Royal Institution Mathematics Masterclass The Fibonacci Connection:The series, the number, the string University of Surrey
2015 March 19 University of Surrey Amazing Maths: The Fibonacci Connection:The series, the number, the string
2015 March 14 Royal Institution Mathematics Masterclass Forgotten Fractions University of Surrey
2015 March 2 IMA NW Annual Sixth Form Lecture at Manchester Grammar School Polygonal Numbers, Pictures and Proof
an article on my Festival talk
Fibonacci, Phi, Food and Flowers" and my image chosen as one of the 50 Visions of Mathematics to celebrate the event.
See the image and explanation here
2014 July 14: Puzzles, Pictures and Proofs talk at the Fibonacci Numbers and Applications Conference 16 at Rochester, NY, USA
2014 July 3: IMA 50 Anniversary Festival of Maths and Its Applications, University of Manchester,
the opening talk on Fibonacci, Flowers, Phi, Food and Flowers
2014 March 29: Manchester POPQUIZ talk: Numbers You can Eat!
2014 February 27: Numbers You Can Eat at Manchester College
2013 March 18: SciBAr (British Science Association, Didsbury, Manchester)
Why Nature likes the Fibonacci Numbers
2012 September 20: IMA NW Branch Student Teacher Lecture
on The Fascinating Fibonacci Numbers for beginning PGCE Students at Manchester Metropolitan University
and others.
2011 March 16: A Festival of Fractions, the IMA
Lancs and NW branch Annual Sixth Form lecture at Manchester Grammar School
2011 March 9: Online video conference with maths students at Meridian International School, Prague, Czech Republic
2011 February 9: Lichfield Schools talk on Flowers as Computers or
The Fibonacci Numbers and the Golden Section in Nature and Mathematics, Friary School, Lichfield
2010 8 July: Mathematical Experiments using the Fibonacci Numbers, Keele University Maths Dept (school talk)
2010 15 May: Investigations with the Fibonacci Numbers, talk at the IMA 12th Young Mathematicians Conference, University of Newcastle
2004 October 6: Fibonacci Fibonacci and Phi for Fun: Facts and Fallacies, IMA NW Branch meeting
2002 November 29: BBC Radio 4 "Numbers" series: Phipodcast
2000: Orkney Science Festival: on Patterns and Numbers in Plants
1998: Orkney Science Festival: on Patterns and Numbers in Plants
1998: BBC Radio Scotland: 1998 interview for the Orkney Science Festival talks
1996 March: The Fibonacci web page goes live on the web!
Dr Ron Knott
Please contact me by email at
if you want to enquire about a talk at your school or society or Science Festival suitable for
a general audience or for mathematics students.
listen again online or download the podcast.
Ph.D(1980, University of Nottingham), M.Sc (1976, University of Nottingham), B.Sc (Pure Maths, University of Wales),
C.Math, FIMA, C.Eng, MBCS, CITP
A Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates
and all papers,
conference report, list of attendees and Problems Section
are now available online.