1480.41 – Broken $\surd$ Key

Use a guess and check method. Here's a good one. Start with a guess $G$ whose square is close to $173$. Average $G$ and $173/G$, that is, calculate $(G + 173/G)/2$. Repeat until you are satisfied.

In our example, take $G = 12$, say, because $12^2 = 144$ isn't that far from $173$. As a first average, we get

$\frac{1}{2} \cdot \left( 12 + \frac{173}{12} \right) = 13.20833333.$

This is our new guess. How do we know when to be satisfied? We square the current result and compare it with 173. In this case:

$13.20833333^2 = 174.46006946.$

which is not yet correct at the thousandth place. So we repeat:

$\frac{1}{2} \cdot \left( 13.20833333 + \frac{173}{13.20833333} \right) = 13.15306257.$

And we check:

$13.15306257^2 = 173.0030549.$

Much better, but still not correct to the nearest thousandth. Again!

$\frac{1}{2} \cdot \left( 13.15306257 + \frac{173}{13.15306257} \right) = 13.15294644.$

This time:

$13.15294644^2 = 173.000000.$

That is, our current result is correct within the nearest $10^{-6}$ or so.

Note: For "the square root of 2", we can more easily say "radical 2". Why is this OK? Well, the Latin word for root is radix. What's a vegetable that comes to mind? Right: radish. So "extracting the square root of 2" is like pulling a radish. Sort of. Anyhow, it's a lot more fun to say "radical 2" than "the square root of 2". More: radical surgery goes to the root of the problem. And so on.