Professor Jim Anderson

Many of us have a favourite number, I am very fond of 259, I suspect that if you were also born in the 1950s you will be able to guess why I like it*.

Professor Jim Anderson from Southampton University School of Mathematical Science came to talk to us about some of his favourite numbers. While these were numbers that he clearly has an affection for - he told us that he has “a soft spot in his heart” for his first choice – he also chose them because he finds them very interesting and because they play an important part in all our lives and indeed in the Universe.

e ( 2.718281828459045235360……………. )

e is sometimes called Euler’s number, after the extraordinarily productive 18th Century Swiss Mathematician. It is a transcendental number which means that it is one of those numbers with a decimal part that simply continues for ever without ever settling down into any sort of pattern. e is one of those numbers that keeps popping up in seemingly unrelated situations and is especially associated with the description of natural phenomena, for example radioactive decay, compound interest, and in Statistics, notably in the Normal Distribution – the “Bell Curve” that characterises so many features of our world. It is also fundamentally connected with Calculus and scientific calculators, even the one on your phone if you turn it round to landscape, have a button just for it.

The Golden Ratio φ (1.6180339887……)

The golden ratio is embedded not only in science but also in art. It has been known and loved certainly since the Ancient Greeks and has been explicitly used in architectural design over the centuries. Golden ratio proportions can be seen both in works of art and more everyday items such as postcards and playing cards.

It has a rather puzzling connection with the famous Fibonacci Sequence, this is the sequence that begins 1, 1, 2, 3, 5, 8, 13, 21, 34 ………..where each term is the sum of the previous two terms. (If you divide each term by its predecessor you get closer and closer to φ, (try 34/21 which is already pretty close). This connection is involved in the drawing of a beautiful spiral that we can find in the natural world.

Although φ does have an infinite decimal expansion it is not a transcendental number. Because (unlike e) it is the solution to a polynomial equation, () and we can therefore give its exact value, (), it is called an algebraic number.

𝞹 Pi 3.141592653589793238462643383279502884197………….

Probably the best known of Jim’s numbers, it is the one we remember from school. 𝞹 is not the solution to any polynomial equation and although we have lots of approximations for it (including 22/7 which is actually close enough for pretty much all household purposes) we do not have a way of expressing it exactly. Like e it is a transcendental number.

We all remember that 𝞹 has something to do with circles (it has a massive amount to do with circles, and spheres and roundness in general), but there is much more to it than that. It just keeps appearing everywhere and all the time. Jim’s description was “You turn a corner and you run into 𝞹). Why for instance should the infinite sum add up to ?

𝞹 has apparently now been calculated to over 62 trillion places, no there isn’t much point to this, though it is said to be a way of checking the processing speed of computers, but there is still awful lot we don’t know about it.

Jim’s particular area of study is Non-Euclidean Geometry. As an example of the ubiquity of[LF1] 𝞹 he said that in fact in hyperbolic geometry the largest size a triangle can be is 𝞹. I think he may have regretted saying this because when asked what this means he said that it was definitely something that would require a whole talk to itself.

i the square root of -1

i is the basic building block of the rather playfully named imaginary numbers. Originally almost a game that mathematicians were playing to see what the consequences might be, imaginary numbers are today an essential part of various areas of physics including electronics, fluid dynamics and computer science and it is also fundamental to engineering (in which i is somewhat confusingly called j).

The only thing we know about i, in fact its sole definition, is that i2 = -1. Euler again played a large part in this branch of mathematics and there is a very famous equation, known as Euler’s Identity which sets out the connection between i, e, , 1 and 0.

Constants of Nature and Metrology

In the last section of his lecture Professor Anderson spoke a little bit about some of the important constants that govern our universe and our systems of measurement.

Two of these – the speed of light and Planck’s constant (h) are exact values, respectively 299792458 metres per second and 6.62607015 x 10-34.. It may seem remarkable that we can know exact values for these but the reason is that these days our definitions of metres and seconds are in fact derived from the speed of light in a vacuum. Their rather “random” looking values come about because we didn’t want such basic units to change size because of the new definitions.

The other two constants he picked out were the mass of an electron, approximately 9.10938 x1031 kg

And the proton-electron mass ratio which is approximately 1836.15267 ( remarkably, this is approximately 6).

It is because of the value that these constants happen(?) to have, that our Universe is as it is and the things in it behave as they behave.

Jim Anderson is a science fiction fan, he left us with the thought that these constants could be different and that even a small difference could mean a profoundly changed Universe. Indeed Multiverse Theories propose that such universes do in fact exist, totally beyond our reach, but not beyond our imagination.

*259 was my childhood telephone number.

Louise Furre

June 2022

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