Why do we rationalize the denominator?
It comes down to what a fraction actually means. It means the division of a finite quantity into a countable number of pieces. Let's take for example, , a quantity that appears often in trigonometry. We can rationalise the denominator to give , but why would we want to do that?
We need to think about what is actually meant by a fraction, and that is to divide a finite length into a COUNTABLE number of pieces. When written in the form , it is asking you to divide 1 into pieces. This doesn't make sense conceptually, because we can't count . We CAN however, picture a length of (as the diagonal of a unit square), and we can picture it being divided into 2 pieces. That means makes "fractional sense", which is the division of a finite quantity into a countable number of pieces, while does not.
okay but what's the point of making fractional sense of it? I'm not sure but is the primary reason we change into is to make it easier to calculate if you didn't have a calculator, so a relic of the days before calculators were commonplace? If so, would the same reason apply for why we rationalize the denominator?
What is the point in having a fraction if it doesn't make fractional sense?
You are not seeing surds and fractions for what they really are, quantities. They are NOT operations. does NOT mean "find a number that squares to give us 2", it means "the length of a unit square's diagonal". It's a quantity, not an operation. When we deal with numbers (quantities) we like to be exact and concise - i.e. to keep things in the simplest exact form that makes sense.
The main reason for "rationalizing" denominators is that when we add fractions we have to get common denominators. And that is typically easier if there are no square roots in the denominators.
However, there are times when we want to rationalize the numerator.