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## Discovering Mathematical Constants – Get to Know These Enigmatic Critters and Make Them Your Friends

A tourist guide around the three most important numbers in mathematics: pi(3.14…), phi(1.618…) ie(2.718…). Each of these numbers has independent significance 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, apart from their celebrity, is the fact that each one is *irrational*. That is, they continue forever after the decimal point. None is exact and its value can only be approximated. This is different from normal *rational* numbers like 3.1, 1.2398, 23.675, etc. These numbers have a definite end. Also, any rational number can be described by a fraction of two numbers. For example, let’s take the number 0.875. This is 7 divided by 8. Here’s another one: 5.295 is 1059 divided by 200. Because of all this, irrational numbers are not uncommon. Our little trio of constants is in the company of the square root of 2, 3, 99 and many cube and square roots to infinity.

If these intrepid numbers were animals, they would be as different from each other as a mouse, an insect and a T-rex. So let’s take a closer look at each of these adorable and enigmatic creatures; his story, his tricks and his powers.

Pi ( π ) should be familiar to most people from their high school math lessons. A good percentage of the population recognizes the Pi symbol and knows that 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 very distant life. As a result of its connection with circles, it appears in equations that describe waves. The link is to a physics phenomenon called simple harmonic motion (SHM). This can be illustrated with a simple example. Think of a ball on a wave in the sea. As the wave passes, the ball rises and falls. The vertical movement of the ball is tied to the waveform passing underneath it. These wave equations can be quite complex and are not for the faint of heart. In fact, the description of waves is considered so important that a whole branch of mathematics called “harmonic analysis” is devoted to it.

Phi ( φ ), or as it is popularly called the golden ratio, the golden section or the golden mean, is a number with its origin based not only on mathematics but on aesthetics. It has its humble beginnings in this role with the ancient Greeks; and in particular with its architecture. They used it as a ratio of lengths, mostly for triangular masonry placed on 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 proportion to great effect in their paintings and sculptures. Leonardo Da Vinci, Seurat, Raphael and Salvador Dalí are just a few who owe their success to the use of Phi.

Returning to the mathematics of Phi, we find that it is intimately tied to something else: the Fibonacci series. In short, the Fibonacci series is a series of numbers generated starting at zero and adding up to one. The series is developed by continuing to add consecutive numbers. So the next number in the series is 1 (0 + 1). Next is 2 (1 + 1). Next 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 through the series dividing pairs of consecutive numbers you get a single number. The higher up the series, the closer this number gets to the Phi value. Take our series above. 3/2 =1.5 5/3 =1.666 8/5 = 1.6 . . . . 233/144 = 1.618055556

In 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, appear again and again in nature. From the infinitely large to the infinitely small, patterns repeat themselves. In the news we hear about the search for the dodger **God Particle: The Higgs Boson**. Well, if there is such a thing **God’s number** then it’s Phi! It is everywhere and in everything.

‘e’, sometimes called Euler’s number, is as important as Phi and Pi, but has a problem with its PR. Unfortunately, it is not as well known as the other two numbers, probably because it is difficult to understand. Its origins lie in the development of natural logarithms, a subject that need not be worried about. In plain English, “e” has unique mathematical properties. It plays an important role in understanding how things grow and decay. It is therefore 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 mathematical delights, and you’ll see “e” center stage, in the middle of it all.

Hopefully that bit of math didn’t hurt too much. Maybe Pi, Phi and “e” have whetted your appetite to learn more? I hope.

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