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Math and Life

Calculating e by Hand

It’s become a bit of a tradition amongst math personalities on the internet to, on pi day (March 14th), go to ridiculous lengths to calculate pi by hand. Sometimes this means measuring the period of a pi-endulem, sometimes weighing a cardboard cutout of a circle. I’ve even see people do cartwheels. The rules are simple: calculate pi. No calculators allowed. Bonus points for originality and ridiculousness.

This is of course a riot of fun, and I always participate, but I can’t help but feel there’s another mathematical constant left out.

e, defined below, is a heavyweight math constant. Second only to pi, it is one of the most well-know and widely-used (amongst math people anyway).

def limit.PNG

Unfortunately, the world does not celebrate an e-day, so we don’t see too many people calculate e by hand. Let’s change that.

The standard ways to approximate e typically involve limits and/or infinite sums:

def sum.PNG

Unfortunately, preforming this is incredibly tedious and boring, so I put this project on the backburner.

One day, while surfing the /r/math subreddit, I discovered this interesting fact:

Pick a random number between 0 and 1. Repeat until the sum of the numbers picked is >1. On Average, this will take e picks.

This is perfect.


Of course, I could use a computer to generate the random numbers for me. However, that feels a little too close to using a calculator. I’m going to need dice.

The original plan was to get ahold of a d20 (a 20 sided die) then tally up the number of rolls it would take to get at least 21.

After explaining the situation to the man behind the counter at my local comic book store, he recommended something called “percentile dice.” Basically, it’s one d10 numbered 0-9, and another d10 numbered 00-90 (with increments of 10).

 percentile dice reading 25

percentile dice reading 25

These dice are far better, because they gives a more even spread.


Now all that’s left is to sit down for ~2 hours and continually roll and add and roll and add.

Here are the results:

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I decided at the beginning to stop once the 2 column reached the bottom of the page. This way, I can’t subconsciously bias the results (like stop after a run of 3s).

Now we calculate!1

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e ≈ 2.687

Only ~1% error too!


1This was the first time I’ve done long division in a long while

I initially gave the answer as 2.687!2. The 2 was referencing a footnote, and the exclamation point was because math is exciting. Some people (intentionally) misunderstood this to mean 2.687 factorial (non-natural number factorials are a thing apparnetly), squared. This has now been fixed.

pedants.