Why not $1$/irrational?

Question

An older math teacher told me that I shouldn't leave a fraction with an irrational in the denominator. But lately, I keep hearing this from every math teacher that I have ever had.

Thus if I have this fraction

$$\frac1{2^{1/2}}$$

I should always convert it to one that has no irrationals in the denominator

$$\frac{2^{1/2}}{2}$$

But why? My thought are that, just like the fact that we can't write out irrationals, we cant write fractions with irrational as denominators as decimals, but we can approximate them, so I don't see the problem. If we approximate $1/\sqrt{2}$, it would be around 0.70710678118, so I dont understand why I shouldn't include irrationals in denominators, at least if I don't want some points deducted from a test.

Answer 1

If I want to compute $1/\sqrt{2}$, I have to divide by $1.414$, at first glance. Ick! 4-digit long division! But if I convert it to $\sqrt{2}/2$, I just have to divide by $2$. Heck, I can do that in my head.

Answer 2

Much of it is a matter of aesthetics. The radical in the denominator is "ugly."

Part of it is just to teach people the algebra so that they know that they can move the radical around.

However, after you progress to a certain level you will quite frequently see $\frac 1{\sqrt 2}$

Regarding your example... $\frac {1}{2^{\frac 12}}= \frac {2^{\frac 12}}{2}.$ If you are going to use the fractional notation, then you might as well say $2^{-\frac 12}$. I think that is neater. Again it is matter of aesthetics.

Answer 3

It's a convention. That's all.

Obviously, $\frac 1{\sqrt {2}} = $ the number, $x$ so that $x*\sqrt{2} = 1$ DOES exist and it is the same number that $\frac{\sqrt{2}}2$ is. So you can write it as such. They advice then becomes you shouldn't. In which case it's very reasonable to ask why not?

I had a professor who used to complain about it not making any sense. And although converting $\frac{1}{\sqrt {2}}$ can be converted to $\frac {\sqrt{2}}{2}$ straightforwardly (to the frustration of elementary students every where), $\frac 1{\pi}$ can not.

To which we realize the rule isn't about irrationals, but about radicals (roots). But again, why?

I think it comes down to an algebraic perspective. If you are given a mess of terms say $5x^2 - 20x - 108$, it might be easier to "get" what is "going on" if it is simplified as $5(x-7)(x+3) +2$. (Or not. If you are simply doing an engineering situation, you don't care that anything is "going on"; you just want a formula for output from input.) Likewise if you are given $\frac{4 + \sqrt{24}}{\sqrt 6}$ it's not clear there is a simplification, and removing radicals is just a useful habit to simplification. ($\frac{4 + \sqrt{24}}{\sqrt 6} = \frac {4\sqrt{6} + \sqrt{144}}{6} = \frac {4\sqrt{6} + 12}{6} = \frac 23\sqrt{6} + 2$... which also equals $\frac{2\sqrt{2}}{\sqrt{3}} + 2= 2\sqrt{\frac 23} +2= \sqrt{8\frac 23 + 4\sqrt{\frac 23}}$; does it really make sense say one is "better"?)

I don't know. I'm mixed on this.

It's time's like this I like to turn to Humpty Dumpty in "Through the Looking Glass" and say "It's matter of who is to be the master".