Any Requests?


I’m thinking to do a second Mathematics A-To-Z Glossary. For those who missed it, last summer I had a fun string of several weeks in which I picked a mathematical term and explained it to within an inch of its life, or 950 words, whichever came first. I’m curious if there’s anything readers out there would like to see me attempt to explain. So, please, let me know of any requests. All requests must begin with a letter, although numbers might be considered.

Meanwhile since there’s been some golden ratio talk around these parts the last few days, I thought people might like to see this neat Algebra Fact of the Day:

People following up on the tweet pointed out that it’s technically speaking wrong. The idea can be saved, though. You can produce the golden ratio using exactly four 4’s this way:

\phi = \frac{\cdot\left(\sqrt{4} + \sqrt{4! + 4}\right)}{4}

If you’d like to do it with eight 4’s, here’s one approach:

https://twitter.com/pepeluis181/status/692767917310607361

And this brings things back around to how Paul Dirac worked out a way to produce any whole number using exactly four 2’s and the normal arithmetic operations anybody knows.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

46 thoughts on “Any Requests?”

    1. I’m not sure what you mean by theory versus practical mathematics. The energy-mass equivalence does follow, mathematically, from some remarkably simple principles. Those amount to uncontroversial things like the speed of light being a constant, independent of the observer, and that momentum and energy are conserved.

      It is experimentally verified, though. We can, for example, measure the mass of atoms before and after they fuse, or fission, and measure the amount of energy released or absorbed as light in the process. The amounts match up as expected. (That’s not the only test to run, of course, but it’s an easy one to understand.) So the reasoning isn’t just good, but matches what we see in the real world.

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      1. Thanks for your clear explanation. I’m not a scientist. Theory wasn’t the right word, then – I was thinking of empirically verifiable which your 2nd paragraph shows. Are the ‘uncontroversial things in your first paragraph also measurable in the real world?

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        1. OK. Well, these are measurable things, in that experiments give results that are what we would expect from the assumptions, and that are inconsistent with what we’d expect from alternate assumptions. For example, we now assume the speed of light (in a vacuum) to be constant. That followed a century of experimentation that finds it does appear to always be constant, and it’s consistent with tests that look to see if there might be something surprising now that we have a new effect to measure or a new tool to measure with. Assumptions about, for example, the way that velocities have to add together in order for this constant-speed-of-light to work have implications for how, say, moving electric charges will produce magnetic fields, and we see magnetic fields induced by moving electric charges consistently with that.

          We can imagine our current understanding to be incomplete, and that the real world has subtleties we haven’t yet detected. But I’m not aware of any outstanding mysteries that suggest strongly that we’re near that point.

          So, given assumptions that seem straightforward enough, and that match experiment as well as we’re able to measure, physicists and mathematicians are generally inclined to say that these assumptions are correct. Or at least correct enough for the context in which they’re used. This is starting to get into the philosophy of science and the concept of experimental proof and gets, I admit, beyond what I’m competent to discuss with authority.

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    1. Oh, number is easy. Three, for example, is the thing that’s in common among Marx Brothers, blind mice, tricycle wheels, penny operas, and balls in the Midnight Multiball of the pinball game FunHouse. Normal subgroup, now that’s hard.

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    1. There’s some good challenges here! My first reaction was to say I didn’t even know what a Dedekind domain was, although in looking it up I realize that I must have learned of them. I just haven’t thought of one in obviously too long, and I like the chance to learn something just in time to explain it.

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    1. Conjecture may work, yes, and fit neatly against axiom trusting that I use that.

      I’m not sure there is a clear guide to when an unproven idea gets elevated to the status of conjecture. I suspect it would defy any rationally describable process. I mean about getting regarded as a name-worthy conjecture. There’s conjectures in much mathematical literature and those tend to mean the person writing the paper got a hunch that something might be so, but didn’t have the time or ability to prove it and is happy to let someone else try.

      But to be, let’s say, the Stangl Conjecture takes more. I suspect part is that it has to be something that feels likely to be true, and which has some obviously interesting consequence if true (or false). That can’t be all, though. The Collatz Conjecture, as I’ve mentioned, seems to be nothing but an amusing trifle. But then that’s also a conjecture that’s very easy for anyone to understand, and it has some beauty to it. The low importance of it might be balanced by how much fun it seems to be and how everyone can be in on the fun.

      I’ll have to do some more poking around famous conjectures, though, and see if I can better characterize what they have in common.

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  1. Equation or Differential Equation, depending on which letter is still open. I am thinking of the way THE FORMULA is depicted in movies, and I believe that it might imply that anything with an equal sign in it is more like Ohm’s Law – a ‘formula’ you just have to plug numbers into. I am sure you can explain the difference between a simple formula and a differential equation nicely :-) Or use Formula instead if F has not been taken.

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    1. Hm. I may take you up on differential equation, since the first nominee — Dedekind domains — is taxing my imagination. And I’d slid continued fractions over to F … but I will think about whether I can find a way to put Formula in under another letter.

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  2. First things that tumble into my mind: Itô integral, Stratonovitch integral, Kulbach-Leibler divergence, Fisher information, Turing machine, Church’s lemma (is this the correct term in English? And you have ‘C’ already, haven’t you?), grammars (both context sensitive and not), Girsanov transformation (sorry for using ‘G’ twice), filtration (I’d really like a good explanation of this one) (and ‘F’), Banach spaces. Orthogonal. Projection. Distance. Metric. Measure. NP-completeness? Gödel’s theorem? Laws of form (that calculus by George Spencer Brown)?

    Might be too nichey, though. You decide.

    Lots of regards, Jacob.

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    1. Well, wow. I do have a couple of these letters taken already — I’ve got through ‘F’ penciled in, plus a couple such as ‘I’ taken after that. But I’ll try to get as many of these as I can done in a coherent form. It’s going to be an exciting month ahead.

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