Here’s Mathematician KP Hart’s Math Question and Answer for March 14th! In celebration of Pi Day, Here are some questions that have to do with that wonderful number, π.
“Who invents the endless series of numbers after 3.14?”
The short answer: nobody.
The long answer: nobody, because that sequence of digits is already determined. The number π once was defined as the ratio of the length of a circle and the length is a diameter. When, in the 19th century, mathematicians put Mathematical Analysis on firm foundations it became clear that this definition was not the most workable one. In the end, the statement “π is the length of a circle with diameter 1” a theorem.
For practical purposes and in order to do something with formulas involving π it would be nice to have an exact numerical expression for it, say as a fraction, or something with some (square) roots in it, but it was proven in the 19th century that that is not possible. We can, therefore, at best make approximations and because the decimal representation has become the one of choice we look for decimal approximations in particular.
There are many formulas for π but there is, as yet, no formula that produces a desired decimal at once: if you want the thousandth decimal you will have to calculate the preceding 999 decimals as well.
A better question therefore is “Who comes up with a formula for the decimal expansion of π?”
More has been written about π than probably any other number in Mathematics. Its Wikipedia page is a good place to start exploring: π on Wikipedia
The number pi represents the ratio of the diameter and circumference of a circle. In spite of this pi is not a rational number but a real number. How is that possible? What does that signify? If we succeed in counting the number of atoms in the diameter (line) and circumference (arc) then can we express pi in a rational number?
Computers are getting faster and more advanced yet they cannot calculate something as relatively simple as pi. Why is that, does the problem lie with the computers? Or is it the question? Or is it possibly the case that pi can be calculated more logically or easier outside the decimal system.
Two related questions on the mystery that is π.
If we are to count atoms on a circle and on the diameter then we should first agree on the compound we shall use and what the length of the diameter is going to be. Both have their influence on the outcome: if we use marbles with a diameter of 1mm and perform the experiment with circles of diameters of 10m and 100m then we will get two different ratios. Those ratios are mere approximations of the real π. Marbles (atoms) with other diameters will give different approximations. We would also not be working in line with Euclid’s Elements: a circle should have no breadth.
The considerations above indicate that intuition does not get us very far when we try to determine the `true nature’ of π. For that we need proper definitions, both of circles and of arc length. And the latter definition is non-trivial. But we have seen this before and will see it again: without unambiguous definitions we will get nowhere.
Starting from those definitions we prove that a circle with diameter 1 has a length and that length we denote π. Then come the questions: how large is π? Can we express π in terms of previously known constants? Preferably as a quotient of two natural numbers.
That last hope was dashed 1761; proofs of the irrationality of π require some knowledge of integral calculus but a persistent first-year student should be able to follow such an argument.
Why is π not rational? Because that was investigated and determined, not by decree but using a (mathematically sound) argument.
This also explains even the best supercomputers cannot calculate `all’ decimals of π: that sequence shows, for now, no special patterns and it cannot be stored on a finite medium. If there is any nice predictive formula for that sequence of decimals then my guess is that it will be found by humans.
There is some progress: there is an algorithm for the `hexadecimals’ of π but the computation time increases with the position of the digit.
Read all of KP Hart’s math questions here!
About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.
I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.
Follow KP Hart on Twitter here!
(This post was first published on Pi Day, 2016)