What Is So Special About 69! ?

69! is not something that you would encounter often in everyday life. I would guess that the typical person does not even encounter the factorial function frequently in day-to-day life, let alone 69!

However, most recently, a unique set of circumstances led me to an intriguing encounter with 69! In this essay, I share my little adventure with 69! and what I learned through the journey.

Before we get into the adventure, to make the story as accessible as possible, I’d like to quickly cover the factorial function first. If the factorial function is nothing new to you, feel free to skip this initial section.

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What is a Factorial?

A factorial is a function in mathematics that is applied to non-negative integers. When this function is applied to an integer n, it results in a multiplicative series of all positive integers less than or equal to n. It is probably easier to look at a few examples:

The key point to note here is that the number n is multiplied with itself and other integers in decreasing order until 1 is reached.

Since the factorial function is resolved into a multiplicative series, the numbers get really large really fast. For instance, 5! = 120, whereas 15! = 1,307,674,368,000. So, in general, we use hand-held calculators or computers to compute factorials of integers.

Now that we’ve covered the basics of the factorial function, we can get into the story of my little adventure with 69!

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The Pursuit of Euler’s Number

Euler’s number (denoted by ‘e’) is one of the most important constants in all of mathematics. I was pursuing certain unique properties of this constant for some research that I had been compiling.

Among the many impressive things that the legendary mathematician, Leonhard Euler, had contributed to mathematics, was the infinite series to compute ‘e’ using factorials. He even computed the value of e correctly up to 18 decimal places in his work: Introductio in Analysin infinitorum (even though he never mentioned how he did it).

During my research into Euler’s number, I got side-tracked and started goofing around with ‘e’ and my calculator. That directly takes us to the next part of the story.

The Scientific Calculator

Once upon a time, I used to work as an engineer. Engineers are creatures who are equipped with things known as ‘scientific calculators’. These are calculators that are more capable than the normal ones, including some minor programming capabilities. I am the proud owner of the one you see in the image below: a CASIO fx-991ES PLUS. It is a handy little thing.

I must admit that this little device is my ‘go-to’ thing for calculations even if I am working on a computer (that is more powerful and more accurate). You may call it old-fashioned or inefficient (or both!). But there is something magical about a hand-held scientific calculator that just draws me in.

Of course, the moment I realise that my calculator is insufficient for the task at hand, I usually resort to a more powerful/precise computational device.

In the context of Euler’s number, I wanted to see how far I could brute force my fx-991ES PLUS into computing the factorial series. In short, I wanted to see how much farther I could get ahead of Euler by using my mighty hand-held device.

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Computing Euler’s Number Using the Scientific Calculator

I used the summation function to compute the series and started off with small upper limits for n. Below, you can see that the upper limit of 10 already allows us to get the value of ‘e’ correct up to 7 decimal places (note: my calculator uses a generalised ‘x’ instead of ‘n’).

This is good, but we are still far away from Euler who got 18 decimal places right! So, like any responsible person, I jacked the upper limit to ‘99’ to see what happens.

Unfortunately, my calculator returned an error message. What was interesting was it took 19 WHOLE seconds (yes, I timed it!) before it displayed the error message. I was sure that some sort of looped calculation was going on and that I had overshot my device’s upper calculation bound.

I continued playing around with the upper bound until I arrived at the exact limit for the fx-991ES PLUS. It turned out to be none other than 69!

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Why 69! ?

My fx-991ES PLUS took 19 MORE seconds to compute ‘e’ correctly up to 9 decimal places in the end. At this point, I had to concede defeat to Euler. I and my puny hand-held calculator were never a match for the great Leonhard Euler!

When I used ‘70’ as the upper bound for the summation function, I got the error message again. So, I was curious as to why this was happening.

I started computing factorials directly on the device, and it turned out that it returned a valid output for 69! but an error message for 70! What I found interesting here was that the outputs for the factorial calculations were instantaneous (unlike the summation function).

I had initially thought that the limiting error message had to do with the summation function. But it had to do with the value of 69! and 70! respectively. Somehow, 70! seemed to hit some sort of a hard limit for the device. A little more investigation revealed that 69! is just under a Googol, whereas 70! crosses a Googol.

I goofed around a bit more and found out that the highest number that the fx-991ES PLUS can handle is 9.9999999999999*10⁹⁹ (that’s thirteen 9s after the decimal point). This is the reason Euler’s summation function could only be calculated up to 69! using my calculator.

Final Thoughts

Ido not know if modern hand-held scientific calculators are limited by the same number. I am sure that anyone playing around with their calculator would find the respective limits rather quickly (every machine has its limits after all). Finding the limits of a machine was never the point though.

The fact that Euler’s number, 69!, and Googol invited themselves to my little adventure involving my scientific calculator made the journey worthwhile and worth sharing.

As researchers and professionals, when we set out to work, we set expectations on ourselves to stumble upon profound discoveries. But little spurts of (arguably) trivial discoveries like these have their moments too. Such (seemingly) trivial adventures make the otherwise dry and long journey worthwhile!

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Further reading that might interest you: Why Exactly Is Zero Factorial Equal To One? and Why Are Analogue Computers Really On The Rise Again?