Square roots and logarithms without a calculator (Part 7)

I’m in the middle of 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.

Today’s topic — slide rules — not only applies to square roots but also multiplication, division, and raising numbers to any exponent (not just to the 1/2 power). To begin, let’s again go back to a time before the advent of pocket calculators… say, the 1950s.

Nearly all STEM professionals were once proficient in the use of slide rules. I never learned how to use one as a student. As a college professor, I bought a fairly inexpensive one from Slide Rule Universe. If you’ve never seen a slide rule, here’s a picture of a fairly advanced one. There are multiple rows of numbers and a sliding plastic piece that has a thin vertical line, allowing direct correspondence from one row of numbers to another. (The middle rows are on a piece that slides back and forth; this is necessary for doing multiplication and division with a slide rule.)


Let’s repeat the problem from Part 6 and try to find

\log_{10} x = \log_{10} \sqrt{4213} = \log_{10} (4213)^{1/2} = \displaystyle \frac{1}{2} \log_{10} 4213.

We recall that \log_{10} 4213 = 3 + \log_{10} 4.213. The logarithm on the right-hand side can be estimated by looking at a slide rule. Here’s a picture from my slide rule:


The important parts of this picture are the bottom two rows. Note that the thin red line is lined up between 4.2 and 4.25; indeed, the red line is about one-third of way from 4.2 to 4.25. On the bottom row, the thin red line is lined up with 0.626. So we estimate that \log_{10} 4213 \approx 3.626, so that \log_{10} \sqrt{4213} \approx \frac{1}{2} (3.626) = 1.813.

Working the other direction, we must find 10^{1.813} = 10 \times 10^{0.813}. We move the thin red line to a different part of the slide rule:


This time, the thin red line is lined up with 0.813 on the bottom row. On the row above, the red line is lined up almost exactly on 6.5, but perhaps a little to the left of 6.5. So we estimate that 10^{1.813} \approx 64.9 or 64.95.

The correct answer is 64.907\dots.

Not bad for a piece of plastic.

green line

Because taking square roots is so important, many slide rules have lines that simulate a square-root function… without the intermediate step of taking logarithms. Let’s consider again at the above picture, but this time let’s look at the second row from the top. Notice that the thin red line goes between 42 and 42.5 on the second line. (FYI, the line repeats itself to the left, so that the user can tell the difference between 42 and 4.2.) Then looking down to the second line from the bottom, we see that the square root is a little less than 65, as before.

In addition to square roots, my personal slide rule has lines for cube roots, sines, cosines, and tangents. In the past, more expensive slide rules had additional lines for the values of other mathematical functions.

green line

More thoughts on slide rules:

1. Slide rules can be used for multiplication and division; the Slide Rule University website also a good explanation for how this works.

2. In a fairly modern film, Apollo 13 (released in 1995 but set in 1970), engineers using slide rules were shown to dramatic effect.

3. Slide rule apps can be downloaded onto both iPhones and Android smartphones; here’s the one that I use. I personally take great anachronistic pleasure in using a slide rule app on my smartphone.

4. While slide rules have been supplanted by scientific calculators, I do believe that slide rules still have modern pedagogical value. I’ve had many friends tell me that, when they were in school, they were asked to construct their own slide rules from scratch (though not as detailed as professional slide rules). I think this would be a reasonable exploration activity that can still engage today’s students (as well as give them some appreciation for their elders).

3 thoughts on “Square roots and logarithms without a calculator (Part 7)

  1. Hi, I found a typo: “…when they were in school, they can to construct their own slide rules from scratch…”

    Also the link to the slide rule app that you like gave me “not found” error. It seems like it’s not in the Play Store anymore, but I managed to find an apk for it by Googling the info in the link.

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