TITLE: Why rationalize the denominator?
QUESTION [109 upvotes]: In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "there is a bias against roots in the denominator of a fraction". But such fractions are well-defined and I'm failing to see anything wrong with $\frac{1}{\sqrt{2}}$ - in fact, IMO it is simpler than $\frac{\sqrt{2}}{2}$ because 1 is simpler than 2 (or similarly, because the former can trivially be rewritten without a fraction).
So why does this bias against roots in the denominator exist and what is its justification? The only reason I can think of is that the bias is a relic of a time before the reals were understood well enough for mathematicians to be comfortable dividing by irrationals, but I have been unable to find a source to corroborate or contradict this guess.

REPLY [2 votes]: Given $\sqrt{2}\approx1.41421356237$, suppose you're challenged to approximately calculate $1/\sqrt{2}$. Now you will find that $1:1.41421356237=100000000000:141421356237=0.7?$, where the calculation of ? isn't easily done.  Knowing that $1/\sqrt2=\sqrt2/2$ it becomes a piece of cake: $1.41421356237:2=0.707106781185$.