compare which one is bigger?

Here's a slightly different approach:

.

(√2011 - 1) / (√2012 - 1) ??  (√2011 + 1) / (√2012 + 1)

Let's cross multiply

(√2011 - 1)(√2012 + 1) ?? (√2012 - 1) (√2011 + 1)

√(2011 * 2012)  -  √2012 + √2011  - 1  ?? √(2011 * 2012)  - √2011 + √2012 - 1

  -  √2012 + √2011   ??  - √2011 + √2012

 √2011 - √2012  ??  √2012 - √2011

2√2011 ?? 2√2012

√2011 < √2012

And since the right side is greater at the end (and we haven't multiplied or divided by any negatives).....then the original right side was greater, too