I hadn’t used my scientific calculator in a while, but recently had to use it to work through some calculations for an analog-related story. Yes, I could use a suitable app on my PC or smartphone, but the calculator has a certain tangible feel to it, at least for me.
My particular calculator for this effort was a Texas Instruments TI-30XA, purchased around 2004. It was made in Italy. TI-30XA replaced the very similar TI-30X from around 1993, which “died” due to multiple hairline cracks in the conductor within the keyboard’s flexible-interconnect board. It’s actually interesting to see how little has changed through many generations.
There are many enthusiast and museum websites devoted to calculators; the Datamath site is devoted solely to TI units, especially their mass-market scientific calculators, such as mine. Unlike the much-earlier and legendary Hewlett-Packard HP-35 (1972) and Texas Instruments SR-50 (1974), which were targeted at scientists, engineers and engineering students, the TI “30” series was aimed at high-school students and less-technical college students.
As I used my TI-30XA for some modest calculations, I realized it had one big difference from its predecessor: the new one’s decimal point was so small that it was almost invisible (Figure 1). At first, I was annoyed and confused, as the newer unit had larger and more visible digits (good), but smaller and hard-to-see decimal point (not so good).
Figure 1 The older TI-30X and newer TI-30XA are remarkably similar despite their difference in age, except that the newer calculator has the easier-to-read larger digits but the nearly impossible-to-see decimal point, located between the leftmost “2” and “8” in both photos. Source: www.datamath.org
However, I soon realized that having to figure out where the decimal point was should be a good thing, at least to some extent. Why? Because it forced me think if the answer on the display actually made sense. Several times I realized that it didn’t, and that I had keyed in a wrong number or misused a formula. After all, I’m pretty sure I didn’t want a 1,500-Ω current-sense resistor for a 1-A current flow, even if that’s what the display digits say its value should be; hmmmm….maybe that should really be 0.0015 Ω?
Reality is that achieving accuracy and precision in the analog world to 0.1% is a challenge, and reaching 0.01% is an even bigger challenge (yes, I know accuracy and precision are not the same). Those 0.1 and 0.01 percentages are roughly equivalent to three and four digits, respectively; any extra digits displayed by the calculator—as distinct from extra digits used internally to minimize roundoff/truncation errors—are unnecessary and imply precision which is not there.
Both estimating answers and doing a so-called sanity check are engineering traditions and skills. Back in the really old days “BC” (both “before calculators” and “before computers”), the engineer’s ubiquitous calculation tool was the slide rule (Figure 2). This analog computation device—and there are hundreds of websites devoted to its history, use, and collections—supported multiplication, division, logarithms, exponentials, trigonometric functions. and other special functions, but not addition or subtraction.
Figure 2 This high-end slide rule—top image is the front and bottom image is the back—took quite a lot of practice to use properly and provided results to two and sometimes three significant figures. You needed your own decimal point location, but it got the job done for scientists and engineers until electronic calculation machines were developed and the slide rule became obsolete. Source: International Slide Rule Museum
When learning to use the slide rule and becoming fluent with it, you had to learn more than the basic “mechanics,” which were not trivial. You also had to understand your equations and algorithms, estimate the answer, and figure out where the decimal point should go, since the slide rule did not provide this information. These factors worked against simple-minded garbage in/garbage out (GIGO) calculations.
Some users were better at this technique and some were worse, but it did force everyone to think and ask “what’s a sensible answer” along with the calculation being done with the slide rule. You can marvel at its relative lack of precision and apparent crudeness, but bridges were built, ships and planes were designed, and rockets were launched using it as a tool. It’s a real testament to the skill and intuitive sense of those engineers and designers.
The real message is that when doing calculations using any technology, whether a slide rule, calculator or computer, the prudent engineering practice is to also estimate what the answer should be. If there’s a large discrepancy—by a factor or two or even an order of magnitude, depending on the specific design—it’s a good idea to stop and figure out why. It could be an error in data entry, in applying a formula, or even in some basic assumptions you’ve made.
Have you ever been “burned” by a precise answer that was also incorrect by a substantial amount? Did you accept the precision and not worry about accuracy, or did you follow your engineering “gut” and investigate further? Have you ever seen others follow the precise numbers and tried to convince to stop and re-assess the results?
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