When is lR NOT a complete space?

Hey all!

I was wondering... Everywhere I see examples of completes spaces, I also see that $\mathbb{R}$ is complete with the usual metric. That's fine with me, but I was wondering why I see no example of a metric for which $\mathbb{R}$ is not complete.

For those wondering - I was thinking about all this when trying to prove (or disprove) that $\mathbb{R}$ is complete with $d(x,y)=|x^{\frac{1}{3}}-y^{\frac{1}{3}}|$ .