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Discovering Mathematical Constants – Get to Know These Enigmatic Critters and Make Them Your Friends
A tourist 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 independent importance in the scheme of things, from describing the exotic properties of energy to the very fabric of matter, and everything in between!
What they all have in common, besides their fame, is that each is irrational. That is, they continue indefinitely after the decimal point. None are exact and their value can only be approximate. It’s different from normal rational numbers like 3.1, 1.2398, 23.675 etc. These numbers have a definite end. Moreover, any rational number can be described by a fraction of two numbers. Take for example the number 0.875. That’s 7 divided by 8. Here’s another: 5.295 is 1059 divided by 200. However, irrational numbers are not uncommon. Our little trio of constants is in company with the square root of 2, 3, 99 and many cube and quadratic roots at infinity.
If these fearless little numbers were animals, they would be as different from each other as a mouse, a bug, and a T-rex. So let’s take a closer look at each of these adorable and enigmatic creatures; their history, their tricks and their powers.
Pi ( π ) should be familiar to most people from their high school/high school math class. A good percentage of the population recognizes the symbol for Pi and knows that it has to do with the area and circumference of circles. Pi is actually the ratio of a circle’s circumference to its diameter. In the larger world, Pi has a life far removed from that. Because of its connection with circles, it appears in equations describing waves. The link is to a phenomenon in physics called simple harmonic motion (SHM). This can be illustrated with a simple example. Think of a ball on a wave at sea. As the wave passes, the ball rises and falls. The vertical movement of the ball is related to the shape of the wave passing under it. These wave equations can be quite complex and are not for the faint-hearted. In fact the description of waves is considered so important that an entire branch of mathematics is devoted to it called “harmonic analysis”.
Phi ( φ ), or as it is commonly called the golden ratio, the golden section or the golden mean, is a number whose origin is based not only on mathematics but also on aesthetics . He has his humble beginnings in this role with the ancient Greeks; and in particular with their architecture. They used it as a ratio of lengths, mainly for the triangular masonry placed on top of pillars, but it was also incorporated into the dimensions of the buildings themselves. Since then, through the Renaissance, the Industrial Revolution and up to the present day, artists have used rapport to great effect in their paintings and sculptures. Leonado Da Vinci, Seurat, Raphael and Salvador Dali are just a few who owe their success to using Phi.
Going back to the mathematics of Phi, we find that it is intimately related to something else – the Fibonacci series. In short, the Fibonacci series is a series of numbers generated by starting with zero and adding one. The series grows by continuing to add consecutive numbers. Therefore, the next number in the series is 1 (0 + 1). The next one is 2 (1 + 1). The next one 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’s the connection you may ask? Well, if you move along the series by dividing pairs of consecutive numbers, you get a single number. The higher you go in the series, the closer this number gets to 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 relationship between Phi and the Fibonacci series so important? The answer is that these numbers, like Phi itself, occur over and over again in nature. From the infinitely large to the infinitely small, the patterns repeat themselves. We hear in the news about the search for the elusive Divine particle – the Higgs boson. Well, if there is such a thing as the number of god then it’s Phi! He is everywhere and in everything.
‘e’, sometimes called Euler’s number, is just as important as Phi and Pi, but it has a problem with its PR. Unfortunately, it’s not as well known as the other two numbers, probably because it’s hard to understand. Its origins lie in the development of natural logarithms, a subject you need not concern yourself with. Clearly, “e” has unique mathematical properties. It has a big role to play in understanding the growth and decline of things. Therefore, it is used in equations to predict population growth, compound interest, and radioactive decay; but it has more exotic uses. Try to find equations related to quantum physics, string theory, or a host of other cerebral math delights, and you’ll see the “e” center stage, in the middle of it all.
Hope that little bit of math didn’t hurt too much. Could it be that Pi, Phi and ‘e’ just whetted your appetite for more? I hope.
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