Yesterday at my summer internship, I had the pleasure of attending a farewell lunch for a senior manager. Memories were being tossed around with varying affectations of amusement, reverence, or chagrin. But, for some reason, the conversation dwelled on slide rules, ruler-like devices used for mathematical calculations before modern calculators became the norm.
There was total diversity of familiarity represented at the table; some people had been prohibited from using calculators for exams all the way through college and were quite familiar with slide rules, but the other engineering interns and even one full-timer had never even heard of one.
In high school, I had a trigonometry teacher on the verge of retirement, and she made a point to include a lesson about slide rules, perhaps for no other reason than to breathe a little life into double-angle theorems and the like. Wikipedia, though, was definitely necessary to piece together the following description.
It’s easy to understand any addition or subtraction as movement along the uniform number line, but I think most people would visualize multiplication and division as changing the scale of that number line, where the change is dependent on the multiplier. Let’s look at two examples: 2x3 and 2x4. The solutions, in bold, can be found by going to the second mark past zero on the appropriately scaled number line:
A slide rule counters the visualization above by saying that any multiplication or division, like addition or subtraction, is also movement along a uniform scale, except that scale is now logarithmic. Slide rules depend on logarithmic arithmetic wherein multiplication becomes summation, which translates into a physical increment along the logarithmically-scaled ruler. The example 2x3 is handled as log(2x3), which is equivalent to log(2) + log(3), or travel a distance of log(3) from log(2), which you can track on the image below. Start at 2 on the bottom rule, travel a distance of log(3) using the top rule, and you arrive at the correct answer.
The second example, however, can now also be performed on this uniform scale. Again using the image above, start at 2 on the bottom rule, travel a distance of log(4) using the top rule, and the solution again is in line on the bottom rule. Now try 2x2.75. The slide rule with its logarithmic scaling presents an entirely alternative way of understanding multiplication.
There certainly was no nostalgia at the lunch for this predecessor to scientific calculators, but there was a certain element of reverence. The point was made that a multiplicity of understanding can be invaluable in problem solving. I couldn’t agree more. In engineering problems on exams, there are usually a few values given, like the density or viscosity of the material, which must be plugged into the formula at the end to compute the right numerical answer. I’ve often only succeeded in solving those problems, however, because I understand those given values to also be clues as to which formula to start with in the first place.
My sister teaches children with autism, and I immediately wondered, after this lunch, if slide rules could revolutionize alternative learning environments. I wonder how many people who struggle with math could break through the basics to a higher level of understanding. How many people would benefit from a multiplicity of understanding?
There’s a huge difference between learning how to multiply and understanding multiplication, especially in the calculator age. The first will help you find the tip on your restaurant bill, but the second is necessary to carry the concept of scaling into multi-variable, multi-term expressions for real, dynamic systems. Beyond pure math, I think the difference between knowledge and understanding separates mechanics from engineers; people who can expertly maintain and operate machines versus people who can reimagine them.
All of this potential from a technological relic that nobody knows how to use anymore leads me to a bigger question, which I’ll preface by acknowledging that I’m more skeptical about new technology than average. When travelling at the speed of technology, what gets left behind?