Monday, 16 February 2015

My Favourite Mathematicians

Continuing the 'My favourite...' series, I will now describe a few of my favourite past and present mathematicians. Due to the extreme number of extremely good mathematicians, this list was especially hard to form but I think I have narrowed down my favourite five.

Carl Friedrich Gauss
A truly incomparable mathematician who had a major influence in various areas of maths even at a very young age. I cannot name even a fraction of his various contributions to mathematics but they include:

Suffice it to say, he has had an incredible influence in almost all areas of maths since the 18th century.

Euclid, the classic Greek mathematician, was a person who revolutionised maths, introducing and developing whole fields of mathematics, especially geometry (he is known as the 'Father of Geometry') and axiomatic theory.
His most prominent work, the Elements, set the way for maths in the following millenia and until the 21st century was the second most sold work worldwide, following only the Bible.

One of his most important ideas he put into practice was the use of axioms, specifically the 5 Postulates from which all his proofs came:
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

You'll note that the 5th is rather more complex than the other four, but that is also a story for another post...

Leonhard Euler
I don't know whether I can put this better than Laplace when he said:
"Read Euler, read Euler, he is the master of us all."
Euler made great discoveries, like all in this list, in various areas of maths but there is one which I associate most strongly with Euler: Euler's Identity.
Euler's identity is what some, if not most, mathematicians call the greatest work of maths as it links seemingly distinct areas of maths in a wonderfully simple and concise equation:
e^{i \pi} + 1 = 0

  • 'e' is the base of the natural logarithm, fundamental in calculus and discovered by Gauss to be linked strongly with the primes
  • 'π' is the ratio between the circumference and diameter of a circle, fundamental to Euclidean and non-Euclidean geometry (named after Euclid of course)
  • 1 is the multiplicitive identity, i.e. anything multiplied by 1 is itself
  • 0 is the additive identity, i.e. anything add 0 is itself

This simple equation links all of these together and is in my opinion the greatest result in mathematics ever.

Pierre de Fermat
Unlike the others in this list, Fermat was an amateur mathematician but, as E.T. Bell put it, the 'Prince of Amateurs'.
He was most famous probably for not what he did, but what he didn't do. He was famed for claiming he had solved big problems without releasing his proofs, most famously in the case of 'Fermat's Last Theorem':
"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."

The jury is still out as to whether he actually solved it or not, but given that it was only solved 358 years later using areas of mathematics as yet undiscovered, it seems unlikely.
However that does not detract from his brilliance as a mathematician.

Srinivasa Ramanujan

Ramanujan, the most modern mathematician in this list, was an exceptional person. Untrained, self-taught and a bit mad (who isn't?), he was brought over to England by G. H. Hardy, author of A Mathematician's Apology, after the exchange of a number of letters which revealed Ramanujan to be a truly incredible mathematician.
In his comparatively short mathematical life, he came up with around 3900 results notably making progress towards areas of maths including infinite series, continued fractions,...
As Erdős recollected, "Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100."

But what he is now most famous for outside of mathematical circles is the taxicab story, which highlights the brilliance of this man:
"I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

A few others:
  • Blaise Pascal
  • Pythagoras
  • Archimedes
  • Isaac Newton
  • Gottfried Wilhelm Leibniz
  • René Descartes
  • David Hilbert
  • Évariste Galois
  • Kurt Gödel
  • Andrew Wiles

Tuesday, 10 February 2015

My Favourite Maths Books

I have a (relatively) large selection of maths books which I really enjoy reading and which were what I would say sparked my love for maths. I would love to share with you my top 5 of these.

1) Fermat’s Last Theorem by Simon Singh
This is the book that really kick-started my love for maths. It provides a great history of maths from the Greeks to modern days whilst keeping the running theme of Fermat’s Last Theorem. It was the first book to introduce me to the ideas of proof, I read it when I was 11 or 12, and it really left an imprint in my mind about what maths is all about.
It also describes the actual problem extremely well, and shows its difficulty through the vast number of mathematicians attempting it over the 358 years it was unsolved. It is definitely my favourite maths book and will always have a place in my heart. 

2) Alex’s Adventures in Numberland by Alex Bellos
Another book which has shaped my mathematical journey is this one. It contains such a great variety of information, from ancient counting systems to casino probabilities, and shows the applications of maths in everyday life really well. It was definitely a fascinating read and a book I’d recommend to both maths-addicts and people without a maths background alike.

3) The Music of the Primes by Marcus du Sautoy
This books also concentrates on a particular problem, in this case the Riemann Hypothesis, and I think it describes it extremely well in a similar way to Fermat’s Last Theorem in that it gives a deep background of both prime numbers in general and the core question giving a historical context throughout.

Unlike the previous two, it also goes into some quite complex maths which was very interesting and explains it very well.

4) 1089 and All That by David Acheson
1089 and All That, despite being smaller than the others in the list, contains an excellent summing up of what maths is about. Its topics range from the fundamental ideas behind mathematics, “Wonderful Theorems, Beautiful Proofs and Great Application”, to a brilliant finale deriving Euler’s Identity. Definitely a great book for providing insight into all areas of maths and deservedly has a place in my Top 5.

5) The Drunkard’s Walk by Leonard Mlodinow
This book, after which this blog is named, is an excellent description of a more specific topic than the others, randomness and uncertainty. I, for one, really found it interest as it was an area I knew little about before and it, like most of the others, showed how well maths can relate to the real world whilst also being beautiful in itself. Suffice it to say, it is an excellent book which I would recommend to anyone interested in the subject

A few others:
  • A Mathematician’s Apology by G. H. Hardy – An insight into the life of one of history’s greatest mathematicians and a defence of maths for the sake of maths
  • The Joy of X by Steven Strogatz – A history of algebra and more maths besides
  • Number Freak by Derrick Niederman – A run-through of various facts about all the numbers from 0 to 200
  • From 0 to Infinity in 26 Centauries by Chris Waring – A history of mathematics from the Greeks to the Renaissance to the future
  • Perfect Rigour by Masha Gesson – A biography of Perelman, who solved one of the seven Millennium Problems, the Poincaré Conjecture, and turned down the million dollars

I’d love to hear what maths books you like and have inspired you in your mathematical journey…

Sunday, 8 February 2015

What are vampire/perfect/happy/palindromic numbers? - Part 2

Continuing my previous post, we will now look at happy and palindromic numbers...

Happy Numbers
For any number, if you add up the squares of each digit and repeat this you will eventually either reach 1 or start an endlessly repeating loop. Numbers which result in a value of 1 are called Happy Numbers.
For example, 19 is a happy number because 12 + 92 = 82, 82 + 22 = 68, 62 + 82 = 100, 12 + 02 + 02 = 1. As you can see, this ends in 1 and 19 is therefore a happy number.
20, however, is not a happy number as 2+ 02= 4, 4= 16, 1+ 62=37, 32+72=58, 5+82=89, 82+92=145, 12+42+52=42, 42+22=20  which is back where we began. In fact, every non-happy number ends up in this sequence; 20, 4, 16, 37, 58, 89, 145, 42, 20.
The first 10 happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32 and 44. Happy numbers which are also prime are, inventively, called Happy Primes; examples of which are 7 and 19. 

Palindromic Numbers
Palindromic numbers, in the same way as palindromic words or phrases, are numbers which are the same when reversed, e.g. 131.
In themselves they may not be that interesting, there are infinitely many and quite regular, but they can be combined with other sets of numbers to give interesting results. For example Palindromic Primes (palprimes) include 313, 797 and Belphegor's Prime: 1000000000000066600000000000001.

By their nature, all Mersenne and Fermat primes are palindromic when written in binary as they are either a run of '1's or a run of '0's with '1's at both ends. For example 31, 11111, is a Mersenne prime and 17, 10001, is a Fermat prime.
Combining these gives Palindromic Happy Prime Nunbers, the first one being 10150006 + 7426247×1075000 + 1 which has over 150 thousand digits.