Question

Why do we need irrational numbers?

Answer 1

A thought or two...

Explanation:

One of the things you would probably like to be able to do with a number system is to measure things.

Suppose we have a square, the length of one side being equal to the span of my hand. That's about `10` inches.

Immediately we have an issue. My span is actually slightly larger than `10` inches, but I cannot express that as a number without either using a smaller unit of length, or going beyond natural numbers to rational numbers.

Putting that consideration to one side (so to speak), suppose we have a square with each side of length `10` inches.

What is the length of its diagonal?

It is about `14` inches, but not exactly.

The exact figure is `10sqrt(2)` inches, which is not even a rational number, let alone a natural number.