Ask Uncle Colin: Which roots have the greater sum?

Dear Uncle Colin,

Which is larger, $\sqrt{3}+\sqrt{11}$ or $\sqrt{5}+\sqrt{8}$? No calculator!

Roots Are Difficult (I Calculated Anyway, LOL)

Hi, RADICAL, and thanks for your message!

If the Mathematical Ninja was nearby – and who can say, they might be – I would probably work out $\sqrt{11}$, the only one of those roots I don’t know to a couple of decimal places. In fact, you can never be too careful: $1089={33}^2$, so $\sqrt{1100}\approx 33+\frac{11}{66}$, or $33.17$. That means $\sqrt{11}\approx 3.32$, which looks about right.

Then 1.73 + 3.32 = 5.05 and 2.24 + 2.83 = 5.07 – so, assuming I’ve rounded right, the second is larger. It’s a bit unsatisfactory, though!

Instead, let’s do it properly: we’ll square both. The first gives $14+2\sqrt{33}$, and the second gives $13+2\sqrt{40}$. I think it’s easier to see if we write those as $14+\sqrt{132}$ and $12+\sqrt{160}$.

Why’s that? I know that ${11.5}^2=132.25$ and that ${12.5}^2=156.25$, so $\sqrt{132}<11.5$ and $\sqrt{160}>12.5$. That means $13+2\sqrt{40}<24.5$ and $14+2\sqrt{33}>24.5$, so the second is larger.

Hope that helps!

- Uncle Colin