I've always assumed the "historical reason" was that when people had to do computations by hand, it was easier to find the decimal value of 1/2^(1/2) [how did

I've always assumed the "historical reason" was that when people had to do computations by hand, it was easier to find the decimal value of 1/2^(1/2) [how did you make those square root symbols?] by dividing 2^(1/2) by 2 than by dividing 1 by 2^(1/2) (assuming that you knew the decimal expansion of 2^(1/2) ). With calculators and computers, that isn't important. Most of the math teachers I know would not penalize anyone for leaving an answer as "1/2^(1/2)". But the idea of rationalizing is important - for example, so that you can express 1/[a + b * 2^(1/2)] in the form p + q * 2^(1/2), where a, b, p, q are rational.