Som's Blog

How I discovered them.

As a child and a student of elementary mathematics, I have always struggled to understand what an irrational number is. To grasp an irrational number, you first need to know what a rational number is. The textbook definition of a rational number is

Any number that can be expressed as p/q, where p and q are integers and q ≠ 0.

For example, 5 is rational, and so is 2.5 since it can be expressed as 5/2. The repeating decimal 3.3333… is also rational, as it is equivalent to 10/3.

And, then what is an irrational number?

A number that is not rational is irrational.

Simple enough, right? Well… not quite. At least, not for me back then. As a child, it felt anything but simple. Those numbers just didn’t make sense. I mean, we were told that Pi is an irrational number — yet its value was given as 3.14. That is nothing but 22/7 or 314/100. And if that’s the case, doesn’t that make Pi rational? My young brain simply refused to believe otherwise.

Then came the eureka moment. I was at a science exhibition in a nearby school. They had a long paper with the numerical value of Pi covering the entire corridor, maybe a thousand decimals or more. I noticed that the digits after the decimal did not form a repeating pattern.

3.141592653589793238462643383279502884197169399375105820974944592307816406286…

It struck me like a bolt.

Any number that can be expressed as a fraction of 2 integers p and q fall into one of these 3 categories

1. q perfectly divides p, as in 10/2=5
2. q divides p so that the decimals do not repeat, such as 5/4 = 1.25
3. q divides p, causing a repeating pattern, such as 4/3=1.333[…3…] or 6/7=0.857142[…857142…]

And there are numbers on the real line that do not fall into the above category, like Pi. The values 22/7 or 3.14 are approximations, not the actual value.

These are irrational numbers!

And there are a lot of them. More than the rational numbers. Yes, sounds unbelievable, but true. The set of rational numbers is countable where as the set of irrational numbers is not.

So next time someone calls you irrational, just smile — you’re in good company with Pi, √2, and an infinite crowd that can’t even be counted!