A tour guide around arguably the three most important numbers in mathematics – pi(3.14…), phi(1.618…) and e(2.718…). Each of these numbers has an independent importance in the scheme of things, from describing the exotic properties of energy, to the very fabric of matter, and everything else in between!
What they all have in common, apart from their celebrity, is the fact that each is irrational. That is they go on forever after the decimal point. None is exact and their value can only be approximated to. This is unlike normal rational numbers like 3.1, 1.2398, 23.675 etc. These numbers have a definite ending. What is more, any rational number can be described by a fraction of two numbers. For example take the number 0.875. This is 7 divided by 8. Here is another one: 5.295 is 1059 divided by 200. For all this, irrational numbers are not rare. Our little trio of constants is in the company of the square root of 2, 3, 99 and many cube and quadratic roots ad infinitum.
If these intrepid little numbers were animals they would be as different to each other as a mouse, an insect and a T-rex. So let us take a closer look at each of these lovable, enigmatic critters; their history, their tricks and their powers.
Pi ( π )should be familiar to most people from their math lessons in High/Secondary school. A good percentage of the population recognises the symbol for Pi, and know it has to do with the area and circumference of circles. In fact Pi is the ratio of the circumference of a circle to its diameter. In the wider world Pi has a life far removed this. As a result of its connection with circles it crops up in equations describing waves. The link is to a phenomenon in physics called simple harmonic motion(S.H.M.). This can be illustrated with a simple example. Think of a ball on a wave at sea. As the wave passes by, the ball goes up and down. The vertical motion of the ball is linked to the wave shape passing under it. These wave equations can be quite complex and are not for the faint hearted. In fact the description of waves is deemed so important that a whole branch of mathematics is devoted to it called ‘harmonic analysis’.
Phi ( φ ), or as it is popularly called the Golden Ratio, Golden Section or Golden Mean, is a number with its origin based not only in mathematics but in aesthetics. It has its humble beginnings in this role with the ancient Greeks; and in particular with their architecture. They used it as a ratio of lengths, mostly for triangular masonry placed atop pillars but it was also incorporated into the dimensions of the buildings themselves. From then on, through the Renaissance, the Industrial Revolution and up to the present day, artists have used the ratio to great effect in their paintings and sculptures. Leonado Da Vinci, Seurat, Raphael and Salvador Dali are but a few who owe their success to using Phi.
Coming back to to the mathematics of Phi, we find it is intimately linked to something else – the Fibonacci Series. Cutting a long story short the Fibonacci Series is a series of numbers generated by starting with zero and adding one. The series develops by continuing to add consecutive numbers. Hence the next number in the series is 1 (0 + 1). The one after is 2 (1 + 1). The one after this is 3 (1 + 2), and so on.
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 . . .
So what is the connection you may ask? Well, if you move along the series dividing pairs of consecutive numbers you obtain a single number. The higher up the series you go the closer this number approaches the value of Phi. Take our series above. 3/2 =1.5 5/3 =1.666 8/5 = 1.6 . . . . 233/144 = 1.618055556
By comparison the actual value of Phi(to 9 decimal places) is 1.618033989 .
So why is this relation between Phi and the Fibonacci series so important? The answer is that these numbers, like Phi itself occur time and time again in nature. From the infinitely large to the infinitesimally small the patterns repeat themselves. We hear in the news about the search for the elusive God Particle – the Higgs Boson. Well if there is such a thing as the God Number then Phi is it! It is everywhere and in everything.
‘e’ , sometimes called Euler’s Number, is every bit as important as Phi and Pi, but it has a problem with its P.R. . Unfortunately it is not as well known as the other two numbers, probably because it is hard to understand. Its origins lie in the development of natural logarithms, a topic you need not concern yourself with. In plain English ‘e’ has unique mathematical properties. It has a large part to play in understanding how things grow and decline. Hence it is used in equations for predicting population growth, compound interest and radioactive decay; but it does have more exotic uses. Try to find equations relating to quantum physics, string theory or a host of other cerebral mathematical delights, and you will see ‘e’ there centre stage, in the midst of it all.
Hopefully this little bit of maths didn’t hurt too much. Could it be that Pi, Phi and ‘e’ may just have wetted your appetite to learn more? I hope so.
