Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$ Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)?
$$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$
Using the calculator I can see that the first one is 63.2455453 and the second one is 63.2455532, but can we tell without touching our calculators?

The answer is: YES, we can!
$$
\begin{align}
(\sqrt{1001}+\sqrt{999})^2&=2000+2\sqrt{1001\times 999}
\\ 
&=2000+2\sqrt{(1000+1)(1000-1)}
\\ 
&=2000+2\sqrt{1000^2-1}
\end{align}
$$
\begin{align}
\text{and that: }(2\sqrt{1000})^2&=4000
\\ 
&=2000+2\sqrt{1000^2}
\end{align}
Since $2000+2\sqrt{1000^2-1}<2000+2\sqrt{1000^2}$
$\therefore \sqrt{1001}+\sqrt{999}<2\sqrt{1000}$
