Man, some things you just can’t un-see.
Exactly. Can’t pin that one on Bluey. For all we know, the horse tried to talk him out of it.
Is there any point in living?
Neigh.
Nope. Blucifer wanted him dead.
Was there a melodica playing in between calls to dance?
Which might remind us that one of the Denver team colors is orange.
This one is mainly for people who tend to be mathematically disinclined;, because I’m sure most people in STEM fields or accounting/finance know this already.
I had a year of calculus in college, and the course content did include exponential functions, so we were introduced to the number e, which is 2.71828… But these were non-rigorous classes intended for non-STEM majors, so they couldn’t really explain what e really means and how it’s derived, much less its importance across all of math, physical science, and engineering,
Fast forward many years. Not too long ago, I became curious about interest rates and how the frequency of compounding affects the effective rate. It’s self-evident that the total return (or interest on a loan) increases with the frequency of compounding, but I didn’t know how high it can go. For all I knew, maybe it was asymptotic and went to absurdly huge amounts if the interest were compounded every nanosecond. I started up the calculator app on my phone and began going through the calculations.
This cheerful English guy explains the result I came up with.
(Video with sound.)
To cut a ten minute video down to a couple of paragraphs, imagine you have an incredibly generous bank that gives you an annual interest rate of 100% on your savings account, not least because you only deposited $1.00. If the interest is calculated every year, then you’ll have $2 the end of the year. If the interest is calculated every six months, you’ll have $2.25.
It’s evident that if you calculate the interest at shorter and shorter intervals, the total amount deposited plus interest will increase. But what I didn’t know was that for this hypothetical deposit of one dollar, the limiting value turns out to be e. If you compound the interest every microsecond or nanosecond, the resulting total will be infinitesimally higher, but it will never be higher than e. In fact it will never quite reach that value.
As it turns out, according to the Wikipedia article, Jacob Bernoulli discovered e while studying compound interest.
This leaves me with two thoughts. First, I wish the people who tried to teach me math had pointed out more interesting results like this. I would have been more interested and motivated in math, and that would have kept the doors open to many career paths which became unavailable to me. And secondly, I have to wonder would Jacob Bernoulli and his mathematical contemporaries could have accomplished with a scientific calculator app.
TIL that if you recruit a bunch of people to move a large, oddly shaped object through a torturous passage - but you prevent them from meaningfully communicating directly with each other (by forbidding speech and gestures, and making them wear surgical masks and sunglasses) - they look an awful lot like ants trying to achieve the same task:
(video is 39 seconds long)
My first exposure to e was from playing around with my scientific calculator, and noticing that there was this button labeled [ln], that seemed to be a logarithm of some sort. By extensive trial and error, I managed to figure out that it was a logarithm base 2.718… (I could have saved myself a lot of effort if I’d noticed what the [2nd] button did to that button). And so I asked my math teacher what the significance was of that number, and why one would want a log of specifically that base.
The answers I got were not entirely satisfactory (for instance, “You can express logarithms and exponentials to any base in terms of that base”, but then, you can do that with any base). So now, when I teach Algebra II and Precalculus students, I try to include a little more of the context about why that base is special. Even then, I feel a little guilty about not telling them all of what’s so special about base e, but then, that’s not really possible until calculus (though I do tell them that when/if they ever do take calculus, they’ll really appreciate base-e exponentials).
I will say that your example of compounded interest is one of the applications that does make it into current textbooks, though. It’s not usually expressed in terms of a hypothetical 100% interest rate, because that’s unrealistic even by textbook standards, but students do learn that A = Pe^{rt} . Of course, this is equivalent to the 100% interest case, if you loo k at longer time intervals, like considering an account earning 1% interest per year to be equivalent to one earning 100% interest per century.
As it happens, my school course in Additional Maths sort of petered out when it came to e, so it’s interesting, all these mumblety-odd years later, to get a bit of clarity as to what it’s for.
In the early 1990s the most popular male name for newborns in France was Kevin, based largely on the popularity of the Home Alone films.
In Germany too, which led to Kevinismus:
e reminds me of another weird number that I thought was traditionally designated τ but I cannot seem to locate it. The definition is τ = 1 / τ - 1; or about 0.618034. I think it may be useful or meaningful in some way, just not sure how.
That’s the Golden Ratio, more traditionally represented \phi. It shows up in a few places, most notably in the Fibonacci Sequence: The ratio of two successive terms of the Fibonacci Sequence (like 5/8 or 8/13) is an approximation to the Golden Ratio.
It was also highly esteemed by the classical Greeks, who considered it to be the most beautiful proportion in artwork. Notably, this only applies to visual art: In music, two notes whose frequencies are in the golden ratio are actually the most dissonant possible.
Also, one runs into the reciprocal of 𝑒, about 0.3678, in such things as [ETA] probability, for example if something has a 1/N probability than the chances of it happening in N trials is 1/𝑒
True, though of course that’s just one more example of e to a power (in this case, e^-1).
That reminds me of how the girl’s name “Madison” took over in the US after the movie Splash came out. It doesn’t appear to have crossed the ocean, though.
e is also enormously important in complex analysis, because of Euler’s formula e^{ix} = \cos x + i \sin x for all x. The Euler’s famous identity e^{i\pi} = -1 is a special case of Euler’s formula when x=\pi.
The name Eric was mostly restricted to Germanic/Scandinavian countries until the book Eric, or Little by Little popularized the name in Britain.
Now it’s spelled with two N’s and a Y, but not where you think.