Michael Barnsley is an English mathematician born in 1946. The code developed by Barnsley is an example of an iterated function system. The code is composed of four affine transformations. of the code is the [more…]
This is also known as the ‘half or triple plus one’ problem. Let be any arbitrary positive integer. If is even, then divide it by two. Otherwise, if is odd, then multiply it by three [more…]
Computer generated graphics have improved enormously over the last forty years. The first computer generated graphic of the Mandelbrot set : A contemporary zooming animation : I particularly like these new images and animations because [more…]
Published by The Random House Publishing Group [US] and Allen Lane [UK] (2007); Penguin Books (2008) ISBN 978-0-141-03459-1 Nassim Nicholas Taleb doesn’t shy away from voicing his opinions, even when they lead to controversy. One [more…]
Western civilisation, and Britain in particular, can be credited with creating the World as we know it. The Industrial Revolution was the transition to new manufacturing processes in the period from about 1760 to sometime [more…]
The numbers exist. No one can own a number. However, some companies/corporations are trying to make it illegal to publish or distribute some numbers on the internet. They are trying to claim that their encryption [more…]
Churchill’s Ministry of Ungentlemanly Warfare: The Mavericks Who Plotted Hitler’s Defeat by Giles Milton Published by John Murray (Publishers) 2016 ISBN 978-1-444-79898-2 Giles Milton is one of those writers and historians with a wonderful story-telling [more…]
The Hat Check Problem, as first described by Montmort in , considers the following problem : A group of people enter a restaurant and check their hats. On leaving, the hat-checker gives the hats back in [more…]
Phil the Test Manager brought up an interesting point on probability the other day. When playing in a bridge contract where you and your partner have eleven of the trump card between you, what are [more…]
“To express remember to memorise a sentence to memorise.” This mathematical constant is a rate of growth. It was discovered by Jacob Bernouilli whilst studying compound interest. Euler referred to the constant as and it [more…]
By Walter F. Smith Sing to the tune of “I Am the Very Model of a Modern Major General” from “The Pirates of Penzance”. If you have to fill a volume with a structure [more…]
Fermat was able to factorise large numbers, such as the above, long before the days of calculators and computers by making use of his little theorem. One could try trial division by primes less than [more…]
Fermat left only one proof. The area of a Pythagorean triangle is never a square number. Fermat wrote , “If the area of a right-angled triangle were a square, there would exist two biquadrates (fourth [more…]
There are trigonometric arguments for interpretation of Plimpton 322 and a number-theoretic argument by Neugebauer.
There is another way of generating the primitive Pythagorean triples using matrices which was discovered by Berggren in 1934 and again by Barning in 1963.
We consider another quadratic Diophantine equation this time in three variables; namely, the Pythagorean Equation : It is called the Pythagorean Equation although the Ancient Egyptians and Babylonians certainly knew how to generate Pythagorean triples. [more…]
As proved by Euler, the value of any infinite continued fraction is an irrational number. Just as every finite continued fraction is a rational number, every infinite continued fraction represents an irrational number. We consider [more…]
A finite continued fraction looks that this : there is an integer part and a nest of fractions. The integer part may be negative, zero or positive. If negative, then it is an improper negative [more…]
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