As a measure, irrationality is irrelevant. When I originally wrote this article, I wanted to somehow prove that irrationality does not exist naturally, that the only way we can possibly create an irrational number is by assuming that infinity exists. What does exist naturally is another question. The intent, however, emerged when I saw that we do explain what these alien numbers are, but not where they come from.
Is infinity an assumption for irrationality?
The golden ratio
The incompleteness of contradiction method
The contradiction method does tell us about why √2 can’t be rational, but not how. If I try to measure the side of that triangle, I would get a rational number. Some will say that that is an approximation, but then how is anything not an approximation? How can you be sure about measuring anything at all?
Can we measure irrational numbers?
We can’t measure anything accurate enough till infinite decimal places. A number is said to be irrational if it can’t be depicted as p/q, where both p and q are integers and q is not 0. It also has to have a never ending non repeating pattern in its decimals.
While you can draw root 2 on the number line, you can’t exactly measure the length of that line segment, or any line segment for that matter. Our measurements are based on approximations to a certain decimal place, and can never reveal the exact value as theorized by mathematics. Therefore we use mathematical theorems to provide an exact value. There are a lot of algorithms to find roots, explained here and here on wikipedia. These methods don’t assume the existence of infinity. Which is the reason why irrational numbers are actually relevant.
The value of root two will come out to be a never ending non-repeating decimal, howsoever we try to calculate it. Similarly, the value of pi, e and the golden ratio will come out to be never ending non repeating decimals as well. This proves that irrationality is a natural phenomenon, and that math does not limit itself. It also proves that the divide-by-zero-error is in fact a mathematical number and a naturally occurring error very relevant to our understanding of mathematics.
I consider this debate (whether irrationality is a natural phenomenon or not) – as a whole – irrelevant, because as long as we stop an algorithm, we get a finite decimal, and as long as we don’t, we know it keeps going on. It is more or less like the Scrodinger’s cat. Infinity does and does not exist. However, as a learning process, I think this debate imparts an important lesson, which is to question the origin of what we study, and learn where it comes from.