Teaching Kids the Beauty of Irrational Numbers

Me: What are rational numbers?

JiMin: Numbers that can be expressed as fractions.

Me: Why do you think we have irrational numbers - numbers that cannot be expressed as fractions?

JiMin: I honestly don’t know. I find it hard to believe that they exist.

Me: Let’s say they weren’t there. What do you think the number line would look like?

JiMin: Hmm. I don’t know. I haven’t plotted irrational numbers on a number line.

Me: OK... we’ll come to that later. First of all, are you convinced that rational numbers when expressed in the decimal form either have a recurring decimal pattern or a terminating pattern?

JiMin: Yes… I haven’t proved it though.

Me: Can you prove it?

JiMin: I don’t know; can I?

Me: It may not be easy. But it will be easy to follow it, once I tell you.

JiMin: OK… let me think a little bit about that one. But on the question of proving that irrational numbers exist, if I can prove that √2 cannot be expressed as a fraction, then haven’t I proven the existence of irrational numbers?

Me: Well, yes. That’s true. So you have probably seen the proof by contradiction at some point. Give it a shot.

The point of this conversation was to show how mysterious and profound, deceptively simple concepts like irrational numbers are. 

Proof that rational numbers must either have terminating decimals or must have repeating decimals

While JiMin is still thinking, the proof is not very hard, once told. You can see that every numerator, when divided by the denominator (say n), leaves a remainder between 0 and n-1. Since these are finite, at some point they must start repeating themselves. Hence, every rational number must either be terminating decimal or a repeating decimal.

Creating Irrational Numbers Artificially

Now the question that we need to answer is if there are numbers that don’t have terminating or repeating decimals. If they can be artificially created or exist naturally, then clearly these cannot be expressed as a fraction and we will call them irrational (this is an unfortunate name for these magnificent numbers). Now let’s think of a number say 0.56566566656666566666… This number has a pattern but it never repeats itself. We basically print a 5 followed by a 6. Every time we add one more 6 than the previous time. This simple algorithm will ensure that there is no repeating pattern here. Now there are infinite ways to create these non-repeating pattern numbers. None of these can be represented as a fraction because they don’t have any repeating decimals. So, these are all irrational numbers. While this is interesting, these numbers still look like artificial constructs. Now let’s see if irrational numbers actually emerge more naturally.

Discovering Natural Irrational Numbers

It is quite incredible that irrationals don't have to be created artificially. Square roots of all non-perfect squares like 2, 3, 5, etc. are irrational. I have known this for a while but have only recently been struck by how amazing this result is. It need not have been this way.

In addition, numbers like π and e are irrational. These are different from square roots of natural numbers as they occur in nature but are not algebraic. π is the ratio between circumference and diameter shared by all circles and e is the base rate of growth shared by all continually growing processes. They are not algebraic because they aren’t roots of any polynomial equation with integer coefficients.

There is nothing irrational about these numbers. I’d have called them mysterious or enigmatic numbers. These numbers are mysterious because it’s very hard to bound/express/capture them but they exist naturally and hence are very real. This is what makes them an integral part of a number line which is meant to be a comprehensive representation of all real numbers. 

How do irrational numbers behave on the number line?

Let’s see where is √2 on a number line. It’s between two rational numbers 1.41 and 1.42. But as you zoom in further you can see that it is between 1.414 and 1.415. The thing is you can see you can keep zooming in and √2 would still be between two rational numbers.

It looks like irrational numbers are the minutest continuum that makes up the fabric of the number line. If there were no real numbers that are irrational numbers then we’d have a number line with infinite small gaps in the number line, with jumps needed from one rational number to the next one.  This is a nuanced point. Rational numbers are infinite in number but each number is finitely expressible. Each irrational number, on the other hand, is infinite in the sense that it cannot be expressed completely on the number line.

Last word

All this is quite surreal even though rational and irrational numbers are both classified as “real numbers”. This discussion with JiMin wasn’t meant to prepare him for any test but just to make him aware of the mysteries that are around us, lurking silently, waiting to be explored.

In many ways you never truly understand numbers and with every step, you get exposed to the depth of your ignorance. It’s no wonder that a mathematician no less than John von Neumann once said, "Young man, in mathematics you don't understand things. You just get used to them.”