*The original article was penned for a the magazine, Teacher, published by the Karnataka chapter of Bharat Gyan Vigyan Samiti. The intended audiences are a combination of high school math teachers and students of the age group 13-15, hence the lack of formal language and rigour. Suggestions and comments are welcome*

What is a rational number? A number which can be expressed in the form ( and being integers) is called a rational number. These numbers follow some interesting properties.

Firstly, between any two rational numbers, there is another rational number.

Try this as an exercise - choose two rational numbers which you feel are close by and try to list a few numbers in between them. You will see that you can grow your list forever. Say you pick and . Lets try to follow a pattern when listing numbers in between them. We know for certain that the average of two numbers in between them. So first, I pick their average . Now I take the average of and - . Now I take the average of and - . And so on. Following this technique, I will be able to keep on listing number, all which are between and . We have just demonstrated that there are infinitely many numbers in between any two rational numbers.

Mathematicians call the set of all rational numbers as a densely ordered set because of this beautiful property.

Now let us introduce real numbers. Continue reading Are rational numbers countable?