which is greater?

Bipolarity

If you want a purely analytical method:

Note that [itex] \sqrt{n} [/itex] is a twice-differentiable function, yielding a second-derivative of [itex] -.25n^{-1.5} [/itex] which is negative for all positive n. Thus the first-derivative of the function decreases monotonically for positive n.

Apply the mean value theorem to the intervals [11,12] and [12,13]. You will get an interesting result which wil give you your answer.

BiP

piercebeatz

You could make it an equation and use the same method... i.e. let root(13)-root(12)+x=root(12)-root(11). If x>0, then the left side is greater. If x<0, the right side is greater

micromass

If the left side is greater, then you can't write an = between the sides.

Am I understanding you wrong?

piercebeatz

let x be the difference between the two. if, in the above scenario, x>0, then the left side must be less than the right, and vice versa.

Mark44

And then the problem becomes determining the sign of x.

tahayassen

Why's that?

Edit: Never-mind. I see why. It's because you square both sides, so when you solve for x, you get |x|=some number.

I would go with Bipolarity's method if you need a proof.