Tom Kando
My previous post began a discussion of some interesting mathematical problems which I want to share with you. I am fully aware that I am a novice in this field. Last time, I wrote about prime numbers. Today, I want to talk about another fascinating topic, namely Pi (π).
You have heard of the expression “squaring the circle,” right? It means doing (or trying to do) the impossible.
In order to construct a square with the same area as a given circle, you must first determine the area of the circle, using the following steps:
1. measure the circle’s circumference.
2. measure the diameter, that is the distance across the circle between the two halfway points of the circumference.
3. determine the radius, which is half the diameter.
4. Square the radius and multiply this number by 3.14 (π).
In other words A (Area) = π x r 2 (r is radius)
Then, you simply compute the square root of the circle’s area. This gives you the length of the square’s side.
Example. You want to square a circle which has a circumference of 1,000 meters.
Using the π approximation of 3.14, The circle’s diameter is therefore about 318.5 meters, and the radius is half of that - 159.25 meters. The circle’s area is therefore 3.14 x 159.25 2 = 79,582 square meters.
The square’s side is the square root of this (√79,582) i.e. approximately 282.1 meters.
For practical purposes, this procedure will do. However, the problem is that π is a transcendental number.
Transcendental numbers are non-algebraic and irrational: They are NOT the solution of an algebraic equation.
They cannot be expressed as the ratio of two numbers. They have a non-terminating decimal.
22/7 is often used as an approximation of π. It is about 3.14. Since it is a fraction, it is a rational number. Like π,, it has an infinite, non-terminating number of decimals.
However,UNLIKE π,, the decimals consist of a six-digit repetend. . For example, if you divide 22 by 7 to 24 decimals, you get 3,142857 142857 142857 142857.
“142857" is called a “repetend,” and it repeats ad infinitum. On the other hand, π cannot be expressed as a fraction. Its exact value cannot be known. Like 22/7, π also has an infinite number of decimals, but there is no “repetend,”
Irrational numbers are not equal to the ratio of any two whole numbers. Their digits neither terminate nor repeat.
π is the best known example of such a number.
Which means that we can never know the EXACT value of π!
Incidentally, the mathematician Georg Cantor proved that there are many more transcendental numbers than there are algebraic numbers, even though there are infinitely many of these latter. In other words, Cantor showed the counter-intuitive fact that not all infinities are equal.
So π is the ratio of a circle’s circumference to its diameter = approximately 3.14. In other words, if you know a circle’s exact circumference and its exact diameter, you divide the former by the latter and you get roughly 3.14... This is a constant, regardless of the size of the circle.
We often equate 3.14 with 22/7 - 22 divided by 7.
Because π, ,unlike 22/7, does not contain the “142857" repetend. it is unpredictable. And of course the digits never end. In this sense, it is impossible to EXACTLY square a circle. Proof of this impossibility was provided in 1882 as a consequence of the Lindemann–Weierstrass theorem, which proves that π is a transcendental number (don't ask me how).
Meanwhile, using a variety of different formulas, mathematicians have forged ever more deeply into π’s decimals. By 2019, Google, with the help of powerful computers of course, had managed to calculate π to 31.4 trillion decimal places.
You may ask, “what’s the relevance of π? And of calculating its decimals into the trillions? Touché. All I can say is that to me, one of the attractions of mathematics is that it is so ABSTRACT. In its purest form, it is not “sullied” by specific objects, be they physical or cultural. A number such as π is a characteristic of the Universe.
© Tom Kando 2022;All Rights Reserved
