American π
For those who write dates in the wrong order the 14th March was Pi Day! This is in homage to probably the best known irrational number and its decimal representation to 3 significant figures (3.14).
As you may know Pi is defined as the ratio of a circle’s circumference to its diameter and, when calculated, it gives a never-ending, never-repeating decimal which some people seem to enjoy memorising digits of. It is an extremely important number and it can be found in many equations throughout science — but I do not believe that it is the only irrational number worth knowing about.
In a mathematical context, if a number is irrational then it cannot be expressed as a ratio of two integers and it is therefore often necessary to notate these numbers with symbols to save spending eternity writing them down. If there infinite nature doesn’t intrigue you enough then it may interest you to know that it possible to prove that the square root of any prime number is irrational or that there’s a link with A4 paper or that nature is riddled with them! So, as nice and familiar as Pi is, let us move on to some of its friends and see what they are and where we can find them.
e
Euler’s number, written as e, is a number with some fascinating properties. For a start it can written as a simple infinite sum:
That is to say if you summed together the inverse of every integer, from zero to infinity, multiplied by each integer lower than itself then you would get Euler’s number! This is actually derived from its most useful property:
Meaning that on a graph of y=ex, the value of y and any point, x, will also give the gradient of the curve at that point. This is found to be the shape of exponential growth and the inverse of exponential decay as found in nature and is used in areas such as nuclear chain reactions, radiometric dating and phases of population growth. Further to this, when paired up with good old Pi, Euler’s number is implicated in naturally occurring Normal distributions used within probability theory:
This is incredible in itself as the area under the curve P(x) is equal to 1 despite the function itself containing three irrational numbers! The relationship between e, π and the imaginary number i=√(-1) is one of the most striking in all of mathematics:
This is known as Euler’s identity and is so famous it even appeared in The Simpsons… twice.
θm
Another less well-known irrational number is the magic angle! Again there are links between this irrational number and others which can be seen from its definition:
I understand that may seem a bit abstract, but fear not — I can use words to describe its origins. Imagine a cube and then picture a diagonal line which runs from one corner of the cube, through the centre and to the opposite corner. The angle made between this line and the edge of the cube is the magic angle! You can find the magic angle on any sheet of A4 (or A1, A2, A3, etc.) paper! Once again it would be angle formed between a line drawn diagonally between two corners and the short edge of the paper.
Particularly important use of the magic angle is within solid-state Nuclear Magnetic Resonance (NMR) Spectroscopy used to identify the chemical structure of a sample. Within NMR, particular interactions between magnetic spins lead to broad, inadequate spectra. However, by making use of Magic Angle Spinning (MAS), which requires the sample to be rotated at high speed about an axis which makes the angle θm with the external magnetic field, these interactions average to zero and valuable information can once again be seen within the spectra!
φ
Finally, let us focus on my favourite irrational number — and it is a rational choice — the golden ratio! The golden ratio is defined as the ratio between a and b such that:
Where a > b. The solution of which is:
As you can clearly see there is a relation to the square-root of five which is irrational itself. However, in my opinion, the most pleasing way in which to write the golden ratio is as an infinite continued fraction which shows off its underlying simplicity:
Or similarly as an infinite surd:
These forms elude to another interesting property of the golden ratio concerning powers of φ:
The real beauty of the golden ratio can be seen within a golden triangle spiral, concerning an isosceles triangle defined by the angle:
The ratio between the lengths of the triangles sides is, as you may have guessed, the golden ratio. It is also unique in being the only triangle whose angles are in a 2:2:1 ratio which, if you’re observant, sees five creeping back into the picture. In fact, for those with a penchant of alternative measurements of angles, 36° can be written in radians as π⁄5. As though that wasn’t enough fives for your liking and then prepared to get all occult as golden triangles appear within pentagrams! Sorry, I appear to have gone off on a tangent… Golden triangle spirals are formed by using the parameters of a golden triangle to form a logarithmic* spiral which, when starting from the corner to the golden angle goes on to bisect the sides of the golden triangle in such a way as to form another golden triangle whereby the process restarts and this continues until you have infinite golden triangles!
*Just to blow your mind; here logarithmic refers to natural logarithms which involve Euler’s number!
Away from pentagrams, the golden ratio crops up in another interesting area of mathematics; fractals. Basic fractals are built from self-repeating patterns in which each iteration is scaled from the previous one. For example:
The value of the scaling factor is important in the presentation of the fractal. Too small a scaling factor would lead to the pattern not covering the whole of space, while a scaling factor too large would lead to overlapping. The optimum? The golden ratio of course! This has implications with Penrose tiling which can be shown to act as a quasicrystal with five-fold symmetry… oh, we’re back to five again.
It also turns out that there are many properties which link the golden ratio to the Fibonacci sequence — where the next number in the series is found from the sum of the previous two (1, 1, 2, 3, 5, 8, 13 …). My favourite such relation was noted by renowned German mathematician and astronomer, Johannes Kepler, who saw that if one takes the ratio between the nth and (n+1)th Fibonacci numbers it is approximately equal to the golden ratio and that this estimation gets better as n increases, or, to go back to the maths lingo:
It is due to this relation that Fibonacci numbers are found within naturally occurring systems, as if the optimum continuous distribution is a golden ratio then a quantised optimised system will contain Fibonacci numbers.
But wait, there’s more; other close approximations of the golden ratio can be found throughout nature including in you! It must be noted that its manifestations are rarely exact as nature is seldom perfect however pretty decent approximations of φ can be found all over you. Let’s take your arm for example; split into sections from shoulder to elbow, elbow to wrist, wrist to knuckles and then on down through your finger joints, measure these sections and find their ratios to find out just how golden you are! Some psychologists have theorised that concepts of beauty have been derived from the brain’s interpretation of the golden ratio, which is why many artists and architects have utilised it in their work (though it should be noted that many claimed occurrences are fallacies).
There is much and more to be read about the uses and implications of all these irrational numbers as well as infinitely many other irrational numbers I have not had the chance to mention. If you’re interested to find out more, the internet is your friend and it will guide on your mathematical journey. Also be sure to find the numberphile channel on YouTube for brilliant videos on many areas of mathematics.
Featured image by Emmanuel Laflamme at quartertofour.net. And yes, it is made up of numbers!
About the Author
Nick Payne graduated from the University of Southampton with a BSc in Physics and has since joined the University of Aberdeen to study for a PhD in Medical Physics (only part of the reason for this was to be called Dr Payne).
