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The finest thing I saw at school was e^(i*pi) = -1. I can't immediately see how to explain that in terms of growth rates. Come to that, it's pretty obvious that pi crops up all over the place in maths that's got nothing to do with circles. Deep, some maths is; deep.

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darieme: You beat me too it. I was going to point out that 2) ignored the most important use of e, pi, or i (in my opinion). But I would argue that the identity does have a bit to do with circles, indirectly. More detail here...

http://en.wikipedia.org/wiki/Euler%27s_identity

When our maths teacher showed us e^(i*pi) = -1 I was so smitten I applauded. He looked round to see whether someone was taking the mick; when he saw it was genuine, he beamed. Looking back, I wish I'd applauded, or thanked, my schoolteachers more often. Mind you, he was an exceptionally good teacher.

Only in my University did I encounter the healthy habit of throwing missiles at lousy lecturers: paper darts, mainly, or - in one extreme case - rotten tomatoes. We also stamped bad lecturers from the room - getting a Victorian lecture theatre's wooden floor bouncing under the attention of a couple of hundred pairs of feet is an exhilarating experience.

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How much money did savers lose because of the artificially low interest rates? Isn't that a cost of QE2?

e^100

How much money did savers (who keep their savings in mutual funds and stocks) lose due to an under-supply of money these past 3 years?

Not as much as they should have.

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It is actually shameful that that thing about e is not well known. Education is full of mysticism-mongering such as this and the phenomenon that Yudkowski calls guessing the teacher's password.

Unfortunately the so called promoters of science, instead of paying their attention to fixing this large class of issues, focus on a miniscule cherry-picked subset of issues such as evolution vs creationism. It is all politics, no one has a decent sense of honesty.

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#5. Maybe I’m biased (because my research is health-related) ... In fact, the best treatments would cost nothing (e.g., the Shangri-La Diet). ... Why is personal science, the main subject of this blog, important? Because it is a way out of this stagnation.

Oh boy, what a kook. A real fine example of kookery, in a classic mold suitable for museums.

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Someone should tell Seth that the Industrial Revolution did so have an obvious reason. Take a culture of competing nations, which has already come upon the scientific method --- whose roots can be traced back to the 12th century Universities --- which already has a lot of free land to settle and is hungry for the energy to do so.

Now give them access to fossil fuels and a compelling reason to use them, and the serious need to get water out of the coal mines they're using ....

Sheesh. "As any fool kin plainly see... I can see!"

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The article on e gives nice examples of exponential growth, but it's not particularly clear in its written definition. This is easy: exponential growth occurs when the growth rate is proportional to how big something is. Rather, the article just says that should be "continually growing", but there are many things that grow continually but do not have exponential growth (fingernails, for example).

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In re: the "explanation" of e --

Good grief!!! I learned about e in the tenth grade. It seemed neither complicated nor difficult, unlike this overly long and verbose article.

If you don't understand limits and convergence, you ain't gonna understand it no matter how many words somebody writes. And if you have even the most rudimentary understanding of those two concepts, then all that other stuff is puffery.

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It seemed neither complicated nor difficult, unlike this overly long and verbose article.

I have to disagree. Learning to *use* e isn't all that tough. However, explaining what 'e' means (which obviously isn't necessary to use it in standard high school curriculum) is pretty rare. I'll ask around my associates, but I'm pretty certain that most of use would use some variant of "the value such that e^x = d e^x / dx".

This was the first time I'd heard this explanation and I find it elegant and intellectually concise. Perhaps not particularly necessary to know, but nice to know.

Tom--"Learning to *use* e" is your terminology, not mine. In my tenth grade algebra class we learned to graph expressions, using a table of values and a corresponding graph on graph paper. We learned the *meaning* of e, not as the statement you give, but rather as the limit of (1 + 1/x)^x as x goes to infinity, and also as (1+x)^(1/x) as x goes to zero. Seeing those two graphs, one that went to an asymptote and the other that approached a singularity, was an eye-opener for understanding limits and convergence, which is a useful thing to master *before* trying to understand calculus.

I believe the approach taken in my high school gave me a better preparation than the one in the article, which seems to me to emphasize how to use e, rather than what it is.

BTW, I am intrigued by your statement that most of your associates would use the calculus definition. It's clearly correct, and probably more rigorous, but I wonder ... did you learn it that way originally, or did you learn it my way first?

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I agree with Tom West. This is a great explanation. i feel like I understand something better today.

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Dear Tyler,
I mostly read this column because you expose me to things I'd otherwise not see.

Linking to Seth Roberts' site makes me question your choice of sources. I know something about medical research and life-expectancy calculation. The understanding of these topics that Dr. Roberts displayed with his post was shallow and self-serving.

I really like the economics and technology discussions on this site.

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I think of pi and e as good examples of how the universe can not be comprehensively explicated by math. As simple as the concepts may be, the numbers grow infinitely grotesque.

Reality ain't too much about numbers. God doesn't play infinitely sided dice.

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Stop telling God what to do. (Bohr)

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Hamilton's piece makes sense as it relates to the Fed increasing overall asset prices.
Now, if we could just get the Fed to treat the unemployed as assets.

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