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Prologue

I remember a dialogue I had with a member of the Mathematics community on Orkut about the concept of real numbers. This was around 4-5 years ago and he was one of the really intelligent members out there in that community and had just joined IIT Bombay. To protect his identity lets call him by the name X. So we have a discussion which goes something like the following:

**Me:** So you have been doing great solving the math problems posed by the community.

**X:** Sort of. I really love solving problems.

**Me:** Looking at the depth of your skills, I suppose you must have some idea of real numbers.

**X:** Oh yes, they are just below the Complex Numbers in the number system hierarchy.

**Me:** Yeah that's fine, but what would you tell about real numbers to a guy who does not know about the complex numbers?

**X:** I would say that the real numbers are "the rational numbers and the irrational numbers" taken together.

**Me:** And then what would say about the irrational numbers?

**X:** Irrationals are just those real numbers which are not rational.

**Me:** If you look carefully at what you said, you will notice that there is a circularity involved and you have not defined any of the terms "irrational" and "real" in context of numbers.

**X:** Yes I guess that you are correct, but I don't know how we can avoid that circularity. We have never been told otherwise about real numbers.