It’s been more than a year now! Someday, I will be frequent here.

Anyways, it’s been a while, I love studying maths – not advanced maths, but that high school level maths – everyone knows! I am at a stage where I feel that whatever I studied during my school was just a bunch of formulas and equations, without even questioning the basic facts. Here’s one:

Why is (-1)(-1) = 1?

Why is it that way defined, can we prove. We just accepted the fact 😀

Okay, that brings to this post’s topic – Maths is indeed fun, but sometimes it questions our beliefs, intuitions and common sense! Let’s consider the system of rational numbers – that we studied in schools – 1/2, 21/7,99/100 etc. On the number line, we can identify each of these numbers. If I am given an interval (or a line segment) [a,b], then I can easily show a rational number q such that a<=q<=b, however small the interval may be! [read that again]

This is the reason, why we say that the **rational numbers are dense** on the line. That implies there may infintely many of them, even if the interval is teeny-tiny! This seems intuitive. But hang-on, here is an interesting thing – Even though rational numbers are dense on the line, what if I show a point on the line that does not collide with any of the infinitely many rational numbers? Seems paradoxical right? Given an interval [1,2] with infinite rational numbers, there is a point $p$ on the interval that manages to avoid contact with any of the rational numbers!

Hold your breath, I chose $p=\sqrt{2}$. It lies in [1,2] which has infinitely many rational numbers but this point evades all of them! Confused! That’s the reason no one believed such numbers exist – from Greeks to even neo-Mathematicians – that questioned their rational thoughts – hence they are irrational numbers!

Source: What is Mathematics?