How can Irrational numbers go on forever>

Question

I understand what irrational numbers are and how they were first proved by the ancient Greeks. My question arsis when thinking of how a length of a line can be the square root of two if by definition the magnitude of root 2 is unending. Yet there is no question that if you draw a square with sides of magnitude one that the diagonal of the square is precisely root 2 long? same question with numbers like 1/3

Answer 1

This is a bit like asking why, when nails are simple and useful, they are so difficult to drive with a screwdriver. This doesn't tell you much about nails; it just tells you that the screwdriver is not well-suited to driving nails.

When you say “go on forever” you are referring to the ‘decimal representation’. The representation of a number is not the number itself, but only one particular way of writing it down. It's a tool for thinking about the number. For $\frac1{10}$ and $\frac18$ it's a tool that is well-suited to the job. For $\frac13$ and $\sqrt 2$, it's less well-suited.

When we write one-eighth as “$0.125$” that's because $$\frac18 = \frac1{10} + \frac2{100} + \frac5{1000}.$$ When we try the same thing with $\frac13$, we can never make it quite add up to $\frac13$ exactly. Is that because $\frac13$ is somehow more complicated than $\frac18$? No, it's just because what we were doing, the tool we were using, is closely tied up with the number $10$—notice the $10, 100, 1000$ on the right-hand side. $\frac13$ happens to be less closely related to the number $10$ than $\frac 18$ is.

But why should we care about $10$? Not for any specially good reason, but only because we wear these meaty bodies with ten meaty prongs sticking out in one place. If we happened to have $12$ prongs instead of $10$ you might be asking why $\frac 15$ went on forever and $\frac13$ didn't. The answer would be the same: $3$ and $12$ happen to be related in a way that $5$ and $12$ aren't, just as $8$ and $10$ are related in a way that $3$ and $10$ aren't.

As you observe, the diagonal of a square is simple. Yes, and we do have a simple way to write it down: “$\sqrt 2$”, which expresses the important, simple property that it does have: if you draw a second square whose side is the diagonal of the first square, the second square will be exactly twice the size of the first square.

The decimal representation for $\sqrt2$ goes on forever, but that doesn't mean $\sqrt2$ isn't simple, it just means that decimal representation is an inconvenient way to write it, a bad tool for the job, because $\sqrt2$ is not closely related to the number $10$ in the particular way that decimal representation demands. The fact that $\sqrt 2$ is hard to write down in this one particular notation has more to do with the notation, and less about the number itself.

I hope this is some help.