# 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?

One 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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