Quatre-vingt-deux and still learning the calculus.

We’re told that learning is life long and while it definitely starts at birth, if not before, and if we’re lucky continues in the school where we are placed, but, when school is over learning doesn’t stop. In my own case learning actually began when I left school. i

In any case learning. as long as we’re alive, never stops, And for me now, at 82 years, learning is as much alive as it ever was. Perhaps even more so because now I have so little time. To no longer learn is to no longer live, and I’m not ready to do that.

But I do find myself thinking about time. Is there time to learn even one or two of the hundreds of things I still want to learn. And all the time I know the answer to be no.

Here’s just one example of these hundred things, the letter e. For most of my life e was only a letter. Then a few years ago I decided to learn the calculus in order to do some tutoring in a local high school and it became essential that I understood what e was all about. (As it turned out I never mastered the calculus well enough to do the tutoring.)

I had done school calculus several times, the first time as a senior at Harvard, meeting a math requirement for medical school, then again at Harvard’s adult ed evening program, and finally at the Waring School where I was teaching.

But it’s only now, and with the help of Sal Kahn of the Kahn Academy that I’m learning the calculus, really learning, while getting my first real sense of math’s great power and beauty perhaps best exemplified by the calculus (Sal doesn’t let me forget this). And I’m learning about e, at least in my free moments away from the garden and the kids.

And I’ve learned that e turns up in many different problem contexts, those involving growth or decay (including compound interest), the statistical “bell curve,” the shape of a hanging cable (or the Gateway Arch in St. Louis), in probability problems, and even in the study of the distribution of prime numbers. And how many more are there?

Probably my first encounter with (I don’t remember the year) was as the base of the natural logarithm. A logarithm can have any positive value as its base, but two of the possible base values, e and 10, are the most common, or the most used. — the one ln(x) meaning the natural log of a number x, and the other, log(x) meaning the “common,” or base 10 log of a number x. And it was only much later in my study of the calculus that I encountered e, most often doing problems with logarithmic or exponential functions.

One thing about e I find fascinating is that there are so many different ways of calculating its value, and not one of these ways ever gives an exact number value because e is irrational, meaning not the ratio of two integers.

  • In this first example the value of (1 + 1/n)n approaches e as n approaches infinity.

graph of (1+1/n)^n

n= 1, 2, 5, 10, 100, 1000, 10,000, 100,000
(1 + 1/n)n   =  2.00000, 2.25000, 2.48832, 2.59374, 2.70481, 2.71692, 2.71815, 2.71827


  • In this second one we find the area under the curve of y=1/x


  • In this third calculation, we have the graph of y=(1+n) raised to the 1/n power.


  • And finally, there is  Newton’s brilliant demonstration of the series expansion for e:


So there are right now, in my 80s, things that I’m learning as if for the first time, as the various ways of calculating e. Why wasn’t I made aware of such fascinating and important ideas earlier when I had the time? How did I miss so much of what one can perhaps best learn when one is young? I guess this is why so many schools fail to educate their students. They fail because they fail to reach the kids, that meaning to interest and arouse the kids to learn on their own, that which is the only valid way to learn.

I’ve always wondered why I’m pretty much alone in respect to my love in particular of ideas. I would call e, and pi, ideas. Among my children, my students, my friends and family there is now probably only my wife who is with me in my life long attempt to learn things. Although I probably lose even her in respect to some of the things I’m doing, going to do, or want to do, —solving the Rubik’s cube in less than 30 seconds, throwing a pot on our recently purchased  wheel, understanding Sal Kahn’s videos on infinite series, understanding infinite series (Newton’s calculation of e)…. all of which I’m struggling with still today.

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