Let’s start this off with a thought experiment. Suppose God has come down and given us a set of drafting tools that can be set to exact quantities. Take the compass and perfectly set the two points to be two units apart. Place the center point down and rotate the graphite point 360 degrees. We will call the center C. From the center point C to the northern point of the circle, say this point is A, draw a line. This will be perfectly one unit long (since the radius of the circle is exactly one unit). Do the same to the western point of the circle, this we call B. Connect A and B with a line. This line AB, by the Pythagorean Theorem, will have a length of square root of two.
Does it really though? Grab a ruler and measure this line. It should be a bit less than one and a half units. If you are using the metric system and choose centimeters as your unit, you’d likely get just a bit more than \(1.4\) centimeters. Choosing the inch as your unit, and with a steady hand and good eyes, you should get somewhere near one and thirteen thirty-seconds of an inch (\(1.40625\) inches). Well, this would make good sense as the square root of two is approximately \(1.414\). Yet, we still haven’t verified that the length is truly the square root of two. Did God really give us a perfect set of tools?
Let’s step it up a notch. We will use a computer numerically controlled measuring device called a CMM, which stands for coordinate-measuring machine. As the name suggests, this device will allow us to define an origin and measure objects precisely within a coordinate system. We lay the paper in the CMM and, after defining an appropriate origin, we trace the line we drew using a precision machined ruby tipped probe. Crucially, this ruby tip will not change shape in any foreseeable temperature changes, so we can count on the fact it will not change our measurements from day to day. Our much more precise tool spits out the number \(1.4145\) units!
This is pretty darn close to the approximation we mentioned earlier, \(1.414\). But, the square root of two is irrational. This \(1.414\) number is an impersonator, a stand in for our convenience. The real number goes on infinitely! So, how does our measurement compare to an even better approximation of the square root of two? Taking our approximation to be \(1.414213562373095\), we see an obvious problem. The number we produced is \(1.4145\), the fourth digit after the decimal is wrong. We aren’t there yet.
Now, imagine that we have an even better method of measurement, something that professional physicists and engineers would spend years devising and building. Some kind of massive arrangement of sensors, stainless steel components, glass viewing ports, and wire bundles. We take our modest little paper drawing and seal it in our world class, multimillion dollar measuring machine. After accommodating for the quantum effects of measuring at this level of precision, the computer spits out a measurement of \(1.4142135623731\) centimeters, or \(0.014142135623731\) meters. This precision is down to the femtometers, smaller than a proton! We must surely have a line that is the length of the square root of two!
Or is it? [Queue Jake Chudnow’s Moon Men] How can you be sure? The only way to know if this is true is to observe it and quantify it. Our super-duper precise measurement is still off. Chasing the tail of this irrational number further past the decimal point we can use an approximation that goes to 65 digits:
\(1.41421356237309504880168872420969807856967187537694807317667973799\)
This is wildly more precise than our measurement and we see that we missed the mark at the 13th digit. Even below the scale of protons we are having difficulty in verifying the length of our perfect little line.
We must now confront a major problem in this pursuit. No matter how complex our machines become, no matter how much time and money we pour into this task, we will always be chasing an infinite tail of the square root of two. In fact, the universe itself will prevent us from going further than \(10^{-33}\) centimeters. This is what is known as the Planck length, which is (theoretically) the maximum level of precision in the universe. We cannot, as of right now, understand how the physical world will behave at lengths less than this. A sounder theory of quantum gravity is necessary, something that has yet to be achieved. Even if it were, there are still 65 digits to the most precise approximation we’ve discussed. So, we must go many many orders of magnitude smaller than the Planck length if we want to verify this length to arbitrary precision.
The physical world cannot accommodate this goal. It is impossible. All measurements, all numbers we have produced thus far have only been rational numbers. That is, they can be expressed simply as a ratio of two whole numbers. They may be highly precise, but their digits terminate. The physical world we live in can only be understood and quantified in terms of rational numbers. You can perform mathematical derivations using irrational numbers, like \(\pi\), Euler’s number (\(e\)), or the Golden Ratio (\(\varphi\)) all day, but there comes a point when verifying these mathematical conclusions (so long as they have physical implications) will need to be done. It is somewhat saddening to think that we will never be able to truly hold or observe an irrational quantity.
I understand that this discussion has been had before and these conclusions are by no means novel or profound. The question of “are numbers real” has been addressed and rehashed by many others. Personally though, having a background in both mathematics and a bit of metrology from mechanical engineering, it is interesting to think that we live in a world where rational numbers are the end all be all. We are, in a way, confined to only know a world made of discrete lengths. Students learning imaginary numbers may become a bit uncomfortable by the idea of “arbitrarily” introducing the imaginary unit, but irrational numbers like \(\pi\), \(e\), and \(\varphi\) are not afforded the same level of mystery. We say, well they are part of the real numbers and have easily understood applications in geometry, trigonometry, and exponential growth/decay (of course the imaginary unit has tons of applications, but most high schoolers will not see them). So, there is no big mystery with the irrational constants. Yet, not to disrespect the logic of derivation, the computations we involve these constants in are a façade masking our actual use of rational approximates. A sort of cover for what goes on inside our calculators, our code, and our measurement devices.
This naturally leads into the question “so what?” That is a good question. At the end of the day, it does not matter whether the value of \(\pi\), or any other irrational, can truly be seen physically. For our purposes, the value of \(3.14159265\) will be plenty to allow us to calculate dimensions to sufficient precision. In fact, we can always find a better approximation to any irrational number we want. We can take advantage of the fact that the rational numbers are dense in the real numbers. This means that if we have any two real numbers, say \(a, b\) where \(a < b\), we can always find a rational number \(\frac{p}{q}\) between them, where \(p, q\) are integers. The digits of the numbers themselves may be finite, but we have an infinite number of approximations to choose from depending on how precise our task demands we be. In effect, if you are not a mathematician, theoretical physicist, or other mathematically inclined professional, you will only ever work with the rational numbers, and this endows them with quite a lot of power. The set of rational numbers, \(\mathbb{Q}\), is the upper bound of our physical knowledge of numbers.