Contents of this page

Dr Ron Knott's web pages on Mathematics:

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

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 Radiolisten 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 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.

Other Recreational Mathematics topics

Whole numbers and how to represent them
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?
Pages on different kinds of Fractions:
Fractions and Decimals
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?

Fractions Calculator
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.
Farey Fractions and trees
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.
Egyptian Fractions
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.
An Introduction to Continued 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)!
Pythagorean Triangles
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.
Angles with simple trig values
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.
A Triangle converter: Cartesian, Trilinear and Barycentric coordinates
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.
Figurate or Polygonal Numbers
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.
What's special about the number 2016?
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.


GCSE Interactive Test
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.
A-level Mathematics Interactive Tests
One page on each of about 120 topics for Mathematics A and AS Level originally developed for MEI.
An interactive Randomiser
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).

Maths Games and Jokes

Got It!
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.
Lock and Roll!
a Yahtzee-type solitaire game where you roll dice to score in various categories.
Herchel's Perpetual Calendar
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!
Math Jokes
"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: ronknott at mac dot com

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.

See this map of the campus
and other info you need is here.

2019 February 19 Launch of the ETC (Encyclopedia of Triangle Centers) easy-access search page
trilinears coords are distances from the three sides barycentric coords are the 3 triangle areas from each side to the point A Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates

This page is designed to link into Clark Kimberling's excellent site:
Encyclopedia of Triangle Centers (ETC)

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
2018 26 January
The Annual joint meeting of the Institute of Mathematics and its Applications NW Branch and MathJam was held at the University in Manchester. I gave a presentation on new results on Number Palindromes
2017 The University of Surrey Open Days in 2017
will again host Dr Ron Knott's popular Mini-Maths talks on:
Friday 30 June,
Saturday 1 July,
Saturday 16 September,
Saturday 14 October
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 in new window 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)
on 2016: a date with mathematics using the OEIS
2016 September 15: IMA North West branch Student Teacher Talk "A Date With Mathematics"
at Manchester Metropolitan University (MMU), Room B4.4 (Building 77 on this PDF map next to the Contact Theatre) at 4:30pm. A talk aimed at new PGCE students but open to all, especially teachers of maths.
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
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
2014 December 18: Florence Nightingale Day
at Lancaster University,
schools talk promoting women to pursue mathematics; 18 December 2014, Numbers you can Eat!
2014 October: Article in IMA journal Mathematics Today Vol 50, No 5, October 2014, pages 242-243 (Contents page)
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
2014 January 20: STEMNetworking LIVE at MOSI
2014 January 15: Numbers you can eat!
Talk and workshop with Year 7 pupils at Hazel Grove High School Stockport
2013 November 18: SciBAr (British Science Association, Didsbury, Manchester) Fun with Fractions!
2013 September 23: STEMNetworking LIVE at MOSI
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
2010 16 March : Flowers as Computers with Fibonacci and Phi, talk to the Lichfield Science and Engineering Society
2009 September: British Association Science Festival 2009, Guildford on Flowers as Computers
2008 22 May at The Eden Project, a talk on Numbers in Plants
2007 26 July at The Eden Project, a talk on Numbers in Plants
2007 29 November : BBC Radio 4 Melvyn Bragg's In Our Time November 29, 2007 when we discussed The Fibonacci Numbers.
Click on the link to Listen Again or to download the 45 minute programme as a podcast.
2007 17 March : Cambridge Science Festival
2006 12 May: Coventry Cathedral's on The Fibonacci Numbers in Dan Brown's "Da Vinci Code"
part of the Da Vinci Code evening
2005 11 July: Fibonacci Numbers and Phi: New and News, talk at TSM 11 Technology for Secondary Mathematics
2004 October 6: Fibonacci Fibonacci and Phi for Fun: Facts and Fallacies, IMA NW Branch meeting
2002 November 29: BBC Radio 4 "Numbers" series: Phi podcast
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 ronknott at mac dot com 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.

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