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Discovering Mathematical Constants – Get to Know These Enigmatic Critters and Make Them Your Friends
A tour guide around the three most important numbers in mathematics – pi (3.14…), phi (1.618…) and e (2.718…). Each of these numbers has an independent significance in the scheme of things, from describing great properties of energy, to the softness of matter, and everything else in between!
What they all have in common, apart from being celebrities, is the fact that they are each ignorant. That’s how they go on forever after the decimal point. None are exact and their value can be approximated. This does not seem normal rational numbers such as 3.1, 1.2398, 23.675 etc. These numbers are final. What’s more, any rational number can be described by a fraction of two numbers. For example, take the number 0.875. Here’s 7 divided by 8. Here’s another: 5.295 is 1059 divided by 200. For all that, odd numbers are rare. Our little trio of candies is in the company of the square root of 2, 3, 99 and many cube and quadratic roots ad infinitum.
If these little numbers were animals they would be as different from each other as a mouse, an insect and a T-rex. So, let’s look at each of these lovers; their stories, 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 knows the symbol for Pi, 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 world far away from this. As a result of its connection with circuits it gives rise to 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 in the ocean. As the wave passes, the ball goes up and down. The vertical movement of the ball is linked to the wave pattern that passes under it. These wave equations can be very complicated and are not for the faint of heart. 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 popularly called the Golden Ratio, Golden Section or Golden Mean, is a number with its origin not only in mathematics but in aesthetics. It had its humble beginnings in this role with the ancient Greeks; and in particular with their architecture. They are used as a ratio of lengths, mostly for triangular masonry placed on top columns but also incorporated into the dimensions of the buildings themselves. Since then, through the Renaissance, the Industrial Revolution and until today, artists have used the element to great effect in their paintings and sculptures. Leonardo Da Vinci, Seurat, Raphael and Salvador Dali are just a few who owe their success to the use of Phi.
Returning to the mathematics of Phi, we see that it is intimately connected to something else – the Fibonacci Series. Cutting the long story short The Fibonacci Series is a sequence of numbers generated by starting with zero and adding one. A series grows by continuing to add consecutive numbers. So the next number in the series is 1 (0 + 1). The next one is 2 (1 + 1). This after that 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 go with the sequence of dividing pairs of consecutive numbers you get one number. The higher the series goes the closer this number is to the value of Phi. Take the 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 important? The answer is that these numbers, like Phi itself occur time and time again in nature. From infinity large to infinity small the patterns repeat themselves. We hear in the news about the search for the elusive God particle – the Higgs Boson. Well if you are looking for something like that 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 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 a unique mathematical property. It has a great role to play in understanding how things develop and build. So it has been used in equations for predicting population growth, population growth and radiation damage; but it has more great uses. Try looking up equations related to quantum physics, string theory or a host of other mental mathematical delights, and you’ll find ‘e’ at center stage, amongst it all.
A little bit of math doesn’t hurt much. Could it be that Pi, Phi and ‘e’ might have just whetted your appetite to learn more? I hope so.
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