Pi Day 3/14/20

Mathematics by Richard Bleil

Happy National Pi Day. Yup, pi, 3.14159265359… It’s the circumference distance around the edge) of a circle with a diameter (the distance across) of one.

In First Kings, Chapter 7, verse 23, it says “He made the sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.” If this is taken as an estimate, it estimates pi to be 3 and has an error (yes, an error in the bible) of about 4.5%. If taken as exact, this defines pi to have a value of exactly three. Republican Congresswoman Martha Roby of Alabama actually submitted legislation in 2011 to make this the legal definition of pi, so much simpler than mucking about with all of those digits and backed by the bible.

Well, simpler it may be, but it is nonetheless wrong. A Babylonian tablet estimated to be from about 1800 B.C. suggested the value of pi to be 3.125. This gives an error of about -0.5%. The ancient Egyptians around 250 B.C. pi to be 3.16 (an error of about 0.6%). It’s interesting to note that the true (at least currently accepted value) is about halfway between these two. In fact, the average of these is 3.1425, which has an error of only 0.03%. That is mighty impressive.

Pi is an irrational number. That means that it cannot be expressed as a simple ratio of rational numbers, in other words, it cannot be expressed as a fraction. For example, the fraction 22/7 is a decent estimate of pi, but it’s not exactly pi. This fraction has the decimal value of 3.142857, which only has an error of 0.04%. This is pretty good, but, not perfect. This proof was first presented by Johann Heinrich Lambert in the 1760’s, and several times since using different mathematical approaches that do not rely on actual calculation. That these proofs don’t need to actually calculate the value of pi is important because to do so could simply mean that the value is not known to enough significant figures yet. However, this brute force approach of proving that pi is a rational number has been attempted by computer scientists for decades as many algorithms continue to run, even today, increasing the number of digits known of pi. In fact, currently pi is known to 2.7 trillion (yes, trillion) digits.

That pi is irrational should, perhaps, not be a surprise. There are several constants in the universe that are actually irrational numbers, but there are an infinite number of irrational numbers between two rational numbers, regardless of how thinly sliced the rational numbers are. See, a rational number is any number that can be expressed as a fraction of integers, n/d, where n is the numerator, and d is the denominator. So, let’s assume two rational numbers, n and d. Two adjacent numbers n would have values n and n+1, or, just for the sake of this discussion, let’s say n+0 and n+1. Because 0 and 1 are both integers, and since we’ve already defined n to be some integer, then n+0 and n+1 are both integers as well, and the fractions (n+0)/d and (n+1)/d are adjacent real numbers, regardless of how large n and d are, including if they are infinitely large. However, because there are an infinite number of decimal places between 0 and 1, then there are an infinite number of real non-integer numbers between n+0 and n+1. This means that between any two rational numbers, there is an infinite number of irrational numbers.

I know, infinity blows my mind as well!

Mathematics is so powerful. Currently, there is a new virus spreading throughout the US, and following the mathematical model of a pandemic with frightening accuracy. With this, we can predict where the virus will hit and help prepare for it by providing supplies and resources where we know they will be needed in a timely fashion. Mathematical models are being used to predict short term weather, but also the effects of global warming, telling us where to expect flooding, and crop shifts. As the weather changes, rainfall can be expected to shift, meaning the temperate and desert zones will also shift. Despite naysayers and climate deniers, these models are already in use by the US military to predict changes that must be anticipated for national defense, and agricultural organizations to predict what the best crops can be expected to grow in any given area based on temperature and rainfall. I myself have developed many mathematical models over the years (and still do, for example, for crime prediction), and invariably, pi ends up somewhere in the equations.

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