Why do we need to rationalize fractions?

Question

Teachers often take off points from students who write 1/sqrt(2) instead of sqrt(2)/2. Why do we need to write it as sqrt(2) / 2 ? Where did that convention come from? Do we need to even do it? Why do teachers care so much?

Answer 1

There is no mathematical reason, it is one of "penmanship."

My hypothesis is that it makes grading easier for them, since requiring this will standardize the answers they get into one form, allowing them to mark the answers that match that form "correct" and those that don't "incorrect" without actually numerically evaluating the answers that may be numerically correct, but didn't match their desired form.

Answer 2

Why do we need to rationalize fractions?

To put it simply, you don't. Not in practical life, anyway. Rationalizing or not, the value of the number is the same. However, there is an advantage of rationalizing. Consider, for example, the number $1/\sqrt{2}$. It is known that $\sqrt{2} \approx 1,4142...$. I'll sort of answer your question with another one: if you want to estimate $1/\sqrt{2}$, what is easier? To think that $$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0,7071...$$ or trying to compute $$\frac{1}{\sqrt{2}}\approx \frac{1}{1,4142...}$$ is some way?