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In my sophomore year of high school chemistry, I learned how to do computations using a slide rule. In my junior year, when taking physics, we put aside the old and used an electronic calculator. It could do the four basic functions, but could also do square roots and even cube roots. There were two in our lab, and they were a bit smaller than a breadbox. (But maybe you don't even know that is!)
In my senior year, I was going to be taking Precalculus, but trying to do the whole year over the summer and the fall semester. Technology was quickly developing smaller and greatly enhanced in their capabilities. In the early summer between my junior and senior years, I bought a handheld calculator that came with a black protective case with a loop, so that we could hang it on our belts like all the other cool kids. It could do all the trigonometric, logarithmic, inverse trig, hyperbolic trig, and more (akin to the modern scientific calculator without the graphing capability). It was SO cool.
Guess what one of the first calculations I tried (after just testing to make sure it worked)? I wanted to know what value of n could cause it blow up when I computed n!. Note that I did not just type both an exclamation mark and a period. Well, yes, I did, but the ! is a postfix operator for the factorial function. Quick aside: if f(n) = value of the factorial function, the postfix notation is n!, which can be confusing. So what is the value of 4! when calculated? It is 4 times 3 times 2 times 1. More generally, n! = n * (n-1) * (n-2) * ... 4 * 3 * 2 * 1. Three examples: 4! = 24, 5! = 120, and 6! = 720. So it grows quickly. Hopefully, you have a little grasp of the factorial function.
Now, recall that I asked: what value of n could cause it blow up when I computed n!. What does "blow up" mean? Every calculator has only so much screen space to show an answer. Early calculators could maybe show 8 or 10 digits across the screen. However, 15! = 1,307,674,368,000. I told you this function grows quickly. To compute this on a calculator with a screen that can show only 10 digits, we might be in trouble. A cheap calculator would display an "E", meaning error. My calculator that I bought that summer would shift to scientific notation, perhaps showing 1.307674 x 10^12 (in some fashion). In this case, the 12 is the exponent of the 10, and calculators of the vintage of mine in this summer topped out, allowing the 12 to only grow to 99, since any reasonable calculations will be smaller than this. After all, 10^99 is a number that is bigger than the number of particles in the universe. So my question was, when did my calculator display an "E" (or some error code), because that means it blew up. Go check out your calculator!
Published 2025-08-25.
If you find any error(s) in the text, please let me know. Thanks. Contact me with errors or comments using hibbardac@gmail. [Back to the top] [About the author, Al]
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