Square roots and logarithms without a calculator (Part 1)

This post begins a series of posts concerning the elementary operation of computing a square root. This is such an elementary operation because nearly every calculator has a \sqrt{~~} button, and so students today are accustomed to quickly getting an answer without giving much thought to (1) what the answer means or (2) what magic the calculator uses to find square roots.

I like to show my future secondary teachers a brief history on this topic… partially to deepen their knowledge about what they likely think is a simple concept, but also to give them a little appreciation for their elders.

To begin, let’s go back to a time before the advent of pocket calculators… say, 1955. (When actually teaching this in class, I find the movie clip to be a great and brief way to get students into the mindset of going back in time.)

How did people in 1955 figure out \sqrt{4213}? After all, plenty of marvelous feats of engineering were made before the advent of calculators. So was this computed back then?

green lineOne rudimentary method is simply by trapping the solution. In other words, let’s try guessing the answer to x^2 = 4213 and see if we get it right.

1. First, the tens digit.

  • 60^2 = 3600. Too small.
  • 70^2 = 4900. Too big.
  • Since 3600 < 4213 < 4900, the answer has to be somewhere between 60 and 70.

2. Next, the ones digit. Since 4213 is about halfway between 3600 and 4900, let’s start by guessing 65.

  • 65^2 = 4225. Too big, but not much too big. So let’s try 64 next, as opposed to 62 or 63.
  • 64^2 = 4096. Too small.
  • So the answer has to be somewhere between 64 and 65.

3. Next, the tenth digit. Since 4213 is so close to 4225, let’s start closer to 65 than to 64.

  • 64.8^2 = 4199.04
  • 64.9^2 = 4212.01
  • We already know that 65.0^2 = 4225
  • So the answer has to be somewhere between 64.9 and 65.

And we keep repeating this procedure, obtaining one digit at a time. (My next guess, for the hundredths digit, would be 64.91 or 64.92.) Back in 1955, all of the above squaring was done by hand, without a calculator. With enough patience, \sqrt{4213} can be obtained to as many digits as required.

I distinctly remember using this procedure, just for the fun of it, when I was 7 or 8 years old (with the help of calculator, however). This exercise was far more cumbersome that simply hitting the \sqrt{~~} button, but it really developed my number sense as a young child, not to mention internalizing the true meaning of what a square root actually was. Little insights like “let’s start closer to 65 than to 64 just don’t come naturally without this kind of trial-and-error practice.

For what it’s worth, the above procedure is the essence of the binary search algorithm (from computer science) or the method of successive bisections (from numerical analysis), with a little human intuition thrown in for good measure.

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