Reasoning on the irrational number Pi

It is probable that there’s no numerical value more known than that used to refer the ratio between the circumference of a circle (or its perimeter), and its diameter (the line that divides a circle into two equal halves). This is the value known as Pi.

Pi unrolled. (angio.net)

This value is quite interesting and has the amazing peculiarity of being a dimensionless number, this means that it doesn’t matter if we’re using the dimensions of a circle formed by a galaxy or by a coin to obtain it, the resulting value for Pi will always be the same. A sequence of digits defined as 3.14159265358979323846, if rounded to the 20th decimal digit.

The ratio was named as Pi in 1706 by the Welsh mathematician William Jones; referring the Greek letter ϖ, considering that “Pi” is the first letter of the Greek words of “perimeter” and “periphery” but till today both terms are used to define this value, which is widely known as it is presented to us in school and is used in diverse calculations, such as the circle’s area (ϖr2), a sphere’s volume (4/3 ϖr3), or a sphere’s surface (4ϖr2), just to mention the basic ones.

Pi is a rational number, part of this know group or family of numbers (yes, numbers have  families, if you recall your school days), along with the natural, and integer numbers; but Pi also belongs to a more exclusive group of number denominated transcendental, these numbers, in addition to being irrational, also have the characteristic of not being derived from an algebraic expression, meaning that we cannot obtain Pi from an equation where a variable’s result is the value of Pi. An example of a non-transcendental but rational number is 1.41421356237, this is an irrational number, yet this can be represented as √2, which is an algebraic expression.

But Pi is not just a simple factor to calculate volumes or surfaces of spherical objects, Pi also has interesting properties and present problems yet to be resolved, here are some examples:

  • The value of Pi seems to be random; as the digits in its sequence show no pattern, and so far, this fact has been holding as more numbers in this sequence are calculated.
  • Pi is considered a never ending random sequence, this means that there’s no point in the sequence where the series starts over with the same initial value 14159265… and so on. An example of an ending random sequence is the one obtained from 22/7, which result in the six-digit non-ending and repeated sequence 3.142857142857142857…
  • Several mathematicians have tried to get this value as a series of infinite continued fractions; cases like the expression from Gottfried Leibnitz is a good approximation ϖ = 4(1/1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11+…); there are multiple sequences like this made by different mathematicians to get this value.
  • Is still uncertain if Pi is a Normal number. A normal number is one where all its digits appear in the same proportion; meaning that, for individual digits, any of them appear one tenth of the times in the sequence. And so far it has passed this test, but we don’t know yet all the digits of Pi.

Well, for what I have commented so far, you might consider this value is just a mathematical curiosity, but that’s far from the truth, the value of Pi is a fundamental factor that’s used in various and diverse areas of science and engineering. Here are some examples:

  • Einstein’s General theory of relativity includes the value of Pi in the constant part on the right side of the expression. This equation seems intimidating, but basically, describes the matter/energy distribution in the right section, and how spacetime is curved on the left.
  • The Electric power, that we get and use every day from our power outlets, is based on alternating current, where the charge is reversed periodically; the mathematical formula that explains how alternate current works includes Pi.
  • Airplanes or vessels traveling long distances through the planet use what is called “great circle navigation”, which provides the shortest distance between two points on the planet, saving fuel and time (and money), Pi is used to determine the distances of these great circle routes.

    Great Circle calculation

    Great Circle calculation. (greatcirclemapper.net)

  • Telecommunications are based on mathematics that “gives the representation of a signal as a function of time, in the frequency domain” (tricky, right? makes me remember my college days!), these are denominated Fourier transforms, and all communication devices use them, so in order for your cell phone to push your Facebook “likes” or your Instagram photos to the internet, they are processed using Fourier transforms by the electronics in your cell phones, to convert this information into signals that are sent to the provider’s cellphone tower through microwaves; in this processing, you can guess it, Pi is involved in the calculations.

In addition to all these applications, there are the aficionados that like to play with the decimal digits of Pi, finding curious combinations. For example, there are four self-located decimals in the first hundred million digits of Pi; wait what? what is a self-locating digit? Well is better to explain it with examples. The first self-locating digit in Pi is the first decimal value 1, (from 3.14…), which is in the first decimal position, the next self-locating decimal value is 16470, starting at the 16,470th position in the sequence. Other interesting sequences are found in this value, as the first eleven digits of another famous transcendental number, the natural logarithm (e). The ascending sequence 0123456789, and the descending sequence 9876543210 are present as well in the value of Pi. And, how many digits of Pi have been calculated? (or generated?), well as today, the record belongs to Nicholas Sze, from Yahoo, that, as a mean to promote the capabilities of cloud computing technology, calculated the first 2 quadrillion digits of Pi (2,000,0000,000,000,0000 digits), using a cluster of one thousand computers.

If you want to look for a particular sequence of the decimal digits of Pi, you can use The Pi-Search Page, where you can type your favorite sequence; like birth dates or plate numbers. In this site, you can also get the first million digits of Pi. In case you want to memorize them and challenge the current champion Akira Haraguchi, who memorized and recited 100,000 digits of Pi ( a feat that took him 16hrs 30 min).

But if you don’t want to dedicate too much effort, yet you want to acknowledge Pi in some way, then you can celebrate the “Pi day”, each March 14th (3.14 get it?), which by the way is the birthday of one of the most famous scientist ever existed; Albert Einstein.

And if you wonder if the image in this post, with all the colored dots, has a meaning; well of course it has, this is a spiral with a color-coded distribution of the first 13,689 digits of ϖ.

Regards Alex-ScienceKindle.

Note: This post was originally published on September, 2017.The note below is still nice to remember though.

PS: This note is off-subject, but I need to mention that today was the end of one of the best NASA and JPL missions;  as the space orbiter Cassini dove into Saturn’s atmosphere. Farewell Cassini and thank you for the amazing legacy.  

Cassini

Cassini, 1997-2007 (www.nasa.gov)

 

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