The Lost Art of Significant Digits

More digits don’t always mean more accuracy

The calculator is “more accurate” than the slide rule. Or is it?

When I entered college in the fall of 1970, I had a nice slide rule (or “slipstick” as some of us called it) that I proudly carried in a leather case to my engineering and chemistry classes. Virtually everyone at North Carolina State University had a slide rule then, but by the time I was a senior, if you didn’t have an electronic calculator, you would have a hard time competing.

Unfortunately, the demise of the slide rule brought with it the demise of “significant digits.” The topic of significant digits was actually taught in many classes because the scales on a slide rule did not allow one to make calculations that would result in three or more digits after a decimal point. As a result, answers to homework or test problems were rendered in “scientific notation” such as 7.13 x 10^3. The electronic calculator would give the answer as 7,132.435674... wow! The calculator is “more accurate” than the slide rule! Or is it? I actually had professors who would deduct points if you gave answers that looked like the latter example and exceeded the proper number of significant digits.

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Comments

A definitely important subject

I also used a slide rule in high school, and this made us set up the problems to include units of measurement as well as powers of 10. We had to add the powers of 10 manually, and were also told that the units had to cancel out to give the correct units for the answer. If they didn't, something was missing. This is a good way to error-proof many science and engineering problems.

Reporting a dozen figures is not a good idea but, on the other hand, there is a huge difference between 99% and 99.3% (30%, or 0.3 out of the 1% tail area) if one is in fact entitled to the figure in the 0.1 place. It is always important to pay attention to exactly what one is rounding off.

don' forget von Neumann

Good article. Another aspect of performing arithmetic with numbers having significant digits is that von Neumann demonstrated that the errors tend to cancel out. That is, it is OK to perform very lengthy (computer) calculations. The number of significant digits is not lost - you also don't gain.