eli5: How were square roots calculated in the first place, before calculators?

Question

eli5: How were square roots calculated in the first place, before calculators?

In: Mathematics

Answer 1

You would look them up in a table. The tables were populated by hand calculation.

For example, what’s the square root of 20? Well, you know it’s going to be more than 4 and less than 5, because 16 is 4² and 25 is 5². So you might try 4.5. So then you would square 4.5 and get 20.25. This tells you that the square root of 20 must be a bit less than 4.5. So you could try 4.4. You square 4.4 and you get 19.36.

So now you know that the square root of 20 must be 4.4xxx. So you might try 4.45. 4.45² is 19.8025, so that’s too low. 4.46² is 19.8916, so that’s too low. 4.47² is 19.9809.

That gives us a second decimal place. Now let’s add another decimal place. 4.475² is 20.025625. 4.474² is 20.016676. 4.472² is 19.998784…

So now by essentially guessing and checking, we’ve gotten three decimal places on the square root of 20. 4.472. We could keep going and keep adding decimal places through the exact same method.