bigideas

blog main page

---

Irrational Numbers And Measurement

3 December 2007 – 8:26 am by  

---

I’ve been meaning to post a layman’s explanation of the irrationality of the square root of 2 for a while, but Meep has just done it in the seventh of her series of short films, Meep’s Math Matters. Her video is not only perfectly comprehensible to someone who doesn’t know any maths, it’s also historically informed and very engaging.

Fractions or ratios of integers — a divided by b, which we’ll write as a/b — can represent almost all the numbers we meet in real life. When we measure something, for instance, we always get an answer that’s a fraction. A length of 2.5cm is five halves of a centimetre: 5/2. If it were exactly 2cm, we’d write 2/1 to represent it as a fraction. Such numbers are called “rational” because they can be expressed using ratios. Mathematicians call the set that includes all rational numbers, positive and negative, of all sizes, Q (which stands for “quotient”).

Even very precise measurements give rational answers, because we’re finite creatures and we can only create or take in a finite amount of information. We can write down a decimal number like 0.65183, but that’s just 65183/10000, a fraction. We can write down recurring decimals, too, but it turns out that they’re fractions as well. The irrational numbers — like the square root of 2 — are all infinite, non-recurring decimals. The famous number π (pi) is irrational, and some people get a kick out of memorising long sequences of its decimal expansion, but nobody can ever know all of π, because it never “settles down” to a repeating pattern.

We can, though, imagine meeting an object whose length was π. Take a circle that’s one metre across. Then π is the length of the circumference of the circle in metres. Likewise, if you have a picture frame that’s one metre square then the length of a diagonal brace would be root 2 metres. So in one sense there’s nothing exotic or esoteric about irrational numbers; they often name quantities that we can easily imagine.

Nevertheless, we’ll never meet a picture frame that’s exactly one metre square, or a circle that’s exactly one metre across, or at least we couldn’t know that we had. That’s because it would take an infinitely precise measurement to tell us that it was one metre, not one metre plus or minus some very tiny quantity. And in physical cases, when you get down to very tiny distances it soon becomes meaningless anyway. Once you’re down to sub-atomic lengths, how can you even tell where the picture frame ends? But to be exactly one metre long it would have to be specific far beyond that, measuring infinitely smaller distances than even the Planck length.

Because mathematics provides infinite precision by default most of the time, it’s easy for us to assume that we have it in real life, where mathematics is so very useful. But we don’t. We only ever work with approximations, with simplified models that are of practical utility, even in the mathematical sciences. A formula might give us an exact speed at which the cannonball will hit the target, but the data going into the formula will be subject to a margin of error when observed, and so will the result.

To go back to an old argument, I think a mistake people make about science in general is thinking it gives us something closer to Truth than this. Science is about making finite observations and making rough predictions. It doesn’t expect its predictions to be precisely right because precision is a practical impossibility.

In the world of pure mathematics, we need both Q and the irrational numbers I (which includes π, root 2 and many others) to make a number system that’s well-behaved. This is called the “real” number system, and is the union of Q and I. It’s written as R, and we’ve met R, and some of its subsets and product spaces, in our current mini-series on topology.

Yet the irrationals do behave a little oddly. For example, there are “more” irrationals than there are rationals, even though there are infinitely many of each. And that’s true even though between any two rationals there’s always an irrational, and vice versa. A consequence of this is that the probability of anything having a rational length, if it could be measured with infinite precision, is zero.

So the real number line isn’t an entirely tame place either. Maybe in the future we’ll do a mini-series on real analysis to describe how these properties, and others, arise naturally from the development of R as the keystone in the construction of the calculus.