Rationalizing a numerator

[Ryn]

Hey guys!

Just joined this board and think it's a great idea. I hope to frequent it a little more seeing as though I'm taking some Math classes. Much to be learned.

Can anyone explain to me why and how you rationalize a fraction? I understand that you need to make it rational, but can't see why and how it happens?

Any help is appreciated!

 

[stapel]

Can anyone explain to me why and how you rationalize a fraction? I understand that you need to make it rational, but can't see why and how it happens?

(Note: I am assuming you mean "rationalizing a denominator"; I've never heard of rationalizing a numerator.)

Explaining "how" denominators are rationalized is a bit beyond what we can provide here, but you can find many useful lessons online for rationalizing radical denominators and rationalizing complex denominators.

The "why" is a bit ticklish. In my experience, now that you're in calculus, your instructor might not, in fact, actually care any more. It's much more of an algebra thing. Kind of like how your grammar-school teacher was horrified by improper fractions like 3/2 (it had to be converted to 1¹/₂), but now that you're in "grown-up math", nobody cares, and impropers are actually usually preferred.

:wink:

Eliz.

 

[galactus]

In calculus it is sometimes necessary to rationalize the numerator.

For instance:

\(\displaystyle \L\\\frac{\sqrt{x+h}-\sqrt{x}}{h}\)

Multiply top and bottom by the conjugate, \(\displaystyle \frac{\sqrt{x+h}+\sqrt{x}}{h}\)

By using property of quotients and difference of 2 squares, we get:

\(\displaystyle \L\\\frac{(\sqrt{x+h})^{2}-(\sqrt{x})^{2}}{h(\sqrt{x+h}+\sqrt{h})}\)

\(\displaystyle \L\\=\frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})}\)

Simplify:

\(\displaystyle \L\\\frac{h}{h(\sqrt{x+h}+\sqrt{h})}\)

Cancel h's:

\(\displaystyle \L\\=\frac{1}{\sqrt{x+h}+\sqrt{x}}\)

Now, as h approaches 0, we have \(\displaystyle \L\\\frac{1}{2\sqrt{x}}\)