How Big Is This Number?

I mentioned in a throwway bit in the article on Goldbach’s Odd Conjecture being (apparently) proven that the number 3^{3^{15}} had been a bound in the conjecture. That is, it was proven in 1939 that numbers larger than that had to obey the conjecture, but that it was unproven for numbers smaller than that. I described it as a number that tekes something like seven million digits to write out in full, that is, in a decimal expansion rather than some powers-of-powers sort of thing.

So let me give it a little attention as a puzzle for people who want to pass a little time doing arithmetic. Am I right to say that 3^{3^{15}} would be a number with about seven million digits?

The obvious way to check is to see what Google comes up with if you put 3^(3^(15)), although that turns out to be Bible quotes. Its calculator gives back Infinity, which here just means “it’s a really, really big number”. My Mac’s calculator function and my copy of Octave agree on that. It’s possible to find a better calculator that gives a meaningful answer, but you can work out roughly how big the number is just by hand, and for that matter, without resorting to anything you have to look up. I promise.

Reblog: Free Harvard Course, Abstract Algebra

Here, Gregory Reese points out a reference to another useful course that might have made my undergraduate life a bit easier were there an Internet to speak of in the early 1990s. (These were primitive days, before Google, before Alta Vista, and when we actually put up with xdvi readers and couldn’t have imagined pdf with its tendency to work and user interface that looks like any thought at all was put into it).

In this case Reese is pointing out the Free Harvard course in Abstract Algebra. Abstract Algebra — it gets called just “Algebra” later on, when we’re not worried that undergraduates will think it’s the thing they did in middle school — is kind of what you get by taking the next set in abstracting middle- and high-school algebra.

One of the things that makes algebra a subject important enough to revolutionize thought and to get into middle- and high-school curriculums is the idea that we can do work with a number — add to it, multiply it, divide by it, raise it to powers, take its logarithm, or so — without necessarily having to know what the number is.

In abstract algebra, we consider the things that we do with numbers, in arithmetic — things like adding them, multiplying them, factoring them — and ask, can we do these things with stuff that isn’t numbers? If we put some thought into what these things are, and what we mean by addition and multiplication and such, it turns out we often can. Abstract Algebra is one of the courses that starts on this trail of doing things that look like arithmetic on things which are not numbers.

My Wholly Undeserved Odd Triumph

While following my own lightly compulsive tracking of the blog’s viewer statistics and wondering why I don’t have more followers or even people getting e-mail notifications (at least I’ve broken 2,222 hits!) I ran across something curious. I can’t swear that it’s still true so I’m not going to link to it, and I don’t want to know if it’s not true. However.

Somehow, one of my tags has become Google’s top hit for the query “christiaan huygens logarithm”. Oh, the post linked to contains the words, don’t doubt that. But something must have got riotously wrong in Google’s page-ranking to put me on top, even above the Encyclopaedia Britannica‘s entry on the subject, and for that matter — rather shockingly to me — above the references for the MacTutor History of Mathematics biography of Huygens. That last is a real shocker, as their biographies, not just of Huygens but of many mathematicians, are rather good and deserving respect. The bunch of us leave Wikipedia in the dust.

I assume it to be some sort of fluke. Possibly it reflects how the link I actually find useful is never the first one in the list of what’s returned, so perhaps they’re padding the results with some technically correct but nonsense filler, and I had the luck of the draw this time. Perhaps not. (I’m only third for “drabble math comic”, and that would at least be plausible.) But I’m amused by it anyway. And I’d like to again say that the MacTutor biographies at the University of Saint Andrews are quite good overall and worth using as reference, and are also the source of my discovery that Wednesday, March 21, is the anniversary of the births of both Jean Baptiste Joseph Fourier (for whom the Fourier Series, Fourier Transform, and Fourier Analysis, all ways of turning complicated problems into easier ones, are named) and of George David Birkhoff (whose ergodic theorem is far too much to explain in a paragraph, but without which almost none of my original mathematics work would have what basis it has). I should give both subjects some discussion. I might yet make Wikipedia.

An Overused Intermediacy

I had wanted to talk about the Intermediate Value Theorem, since it’s one of those little utility theorems that doesn’t draw a lot of attention by itself but does have some wonderful results that depend on it. My context was in explaining just what Chiaroscuro had done when he figured out the fifth root of 1/6000th by guessing at it. I mean, he figured he was guessing at it, but there’s good reasons why this guessing would pay off and why he’d get to an answer near enough the right one.

And I wanted to talk about one of my favorite results of the Intermediate Value Theorem, at least as I remembered it: that at any time of the day or night, there must be at minimum a pair of antipodal sites — locations directly opposite the center of the Earth from one another — which have exactly the same temperature. Or the same humidity. Or the same of any meteorological measurement. I had read this, I was sure, in Richard Courant and Herbert Robbins’s masterpiece of mathematics writing, What Is Mathematics? and went digging about to find it precisely stated, particularly since as I remembered it was possible to get any pair of measurements — say, temperature and humidity together — exactly equal at antipodal sites.

Continue reading “An Overused Intermediacy”

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