So while reading it seems more and more familiar, and I’m trying to figure out if some TV show borrowed the plot or why this seems so familiar. I finally checked my collection more closely and, sure enough, it’s familiar because own it and I’ve read it. (Within the last couple of years, at that. As I’ve said many times, my memory for fiction plots is practically non-existent.)

But here’s the eerie part. One character is a mathematician and in one scene he writes out a formula for what he calls the *“second approximation to the normal distribution”*. That formula is in the book, and our friend *e-to-the-power-of-something* is part of that formula. Specifically, it was a Gaussian. (When I say it’s ubiquitous and appears everywhere, I’m not kidding about that, either.)

See this Wiki article for examples.

]]>Sounds like we have similar approaches to learning something. I need to *understand* — I can’t learn by rules or facts. As you say, it pays off big time later.

I think a *lot* of people got disenchanted with math in school. It can be a very poorly taught subject, sometimes by teachers who don’t grasp or love it themselves. As you’ve found from books, just loving it isn’t enough — that doesn’t carry over to others. It takes connecting it or illuminating it in a compelling and engaging way. (That’s why I love that 3Blue1Brown channel — that’s what the guy does, and he’s really good at it.)

If you wanted to tackle linear algebra again, I *highly* recommend his Essence of linear algebra series. It’s 15 videos ranging from the 4-17 minute range; most of them kind of in the middle of that. A lot of working with the Schrödinger equation involves concepts from linear algebra, eigenvectors and eigenvalues key among them. The series involves a visual understanding of matrix transformations that makes them almost obvious.

(Tomorrow’s post involves a video of his that makes the Fourier transform almost obvious. It’s a really cool way to look at it, and this *e-to-the-power-of-i* business is literally at the heart of it. And as I may have mentioned, the Fourier transform underpins the HUP — position and momentum are conjugate Fourier transforms of each other.)

Exactly true about the math looking less cryptic! More and more that quantum math is starting to look like something I understand, although I’m a long ways from being able to work with it.

]]>I’m usually a foundations guy myself. In school, I was always at an disadvantage early on, because while others were just memorizing heuristics, I was trying to grok the concept at a deeper level. Of course, doing so always eventually translates into advantages later on.

I’m gradually listening my way through Lex Fridman’s interview of Scott Aarsonson (Aaronson links to the Youtube version from a recent blog post). One of the things he discusses is how understanding a few key concepts enables a lot in computer science. I found his take interesting.

The problem is that, with math, I’ve just never felt the bug, never been interested in it for its own sake. And when I attempt to dig too much into it, it awakens anxieties from all the struggles I had in school. Math books always seem to be written by someone who thinks it, in and of itself, is inherently beautiful. Well, I’m sorry to say it isn’t beautiful to me. I need to have it relentlessly mapped back to something useful, or it takes enormous discipline for me to continue.

I’ve struggled even getting through a chapter on linear algebra in a quantum computing book, despite knowing that I need it for the rest of the book, because the author switches into pure math mode, shoveling it without mentioning how any of it maps to the quantum subject. (The reading I did do wasn’t totally in vain. I noticed that some of the notation in quantum physics papers suddenly seemed less cryptic.)

]]>Heh! Part of me wants to protest that math is always about something, but I know what you mean. These two posts (one coming tomorrow) have a direct application to quantum mechanics. If you look at the Wiki page for the Schrödinger equation you don’t have to go very far down the page to find the first formula with *e-to-the-i*, and it reappears constantly throughout the page.

It really is a fundamental building block, and having a grasp of what it means opens a lot of doors in physics. (Not just QM, but optics and sound.) It also open a door to the Fourier transform, which turns out to be another basic building block.

FWIW, I have a major commitment to the idea of foundation knowledge. On a practical level, it’s what allowed me to change modes during a changing career. Good foundations make it easier to build new knowledge. An analogy I like is that, if one knows music and has already learned to play an instrument, learning a new instrument is just a matter of learning how to operate the new thing. The more instruments one plays, the easier new ones become.

Math is one of those foundations that tends to show up in many of my favorite places, so I’ve always wanted to know more, and — fortunately for me — I find math fascinating, so it’s easy and rewarding to pursue.

The downside of foundation knowledge — one reason I think a lot of people lack it — is that the ROI really sucks plus it takes forever to acquire. It’s exactly the reaction you’re expressing to math — *what’s this for and what’s the point?* It’s not very satisfying to say, “Well, maybe it will come in handy later,…” It just takes being fascinated by something I guess. (Or discipline I lack, since otherwise I’d know a lot more about chemistry and biology.)

I think there are a number of things with shallow learning curves — long stretches of effort with little payoff. Music was that way for me long, long ago. Then one reaches an inflection point in the curve and progress skyrockets. Suddenly the pieces makes sense and fit together.

At least that’s how a lot of things have worked for me. Long shallow curve with a steep slope after the light bulb finally goes on. (I’ve gotten the impression my curve is shallower than many in the beginning, but I make up for it later.)

]]>There is also a rule: *N*^{1} = ** N**. (We saw that in how all the simple exponential curves passed through 1.0.)

So for this equality:

It’s just:

*N*^{5} = *N*^{1+1+1+1+1}

*N*^{5} = *N*^{1} × *N*^{1} × *N*^{1} × *N*^{1} × *N*^{1}

*N*^{5} = ** N** ×

**QED**!

The next two are ever so slightly more involved:

For the first one, first consider that, trivially:

*N*^{0+x} = *N*^{x}

And also:

*N*^{0} × *N*^{x} = *N*^{x}

Therefore (dividing both sides by *N*^{x}):

*N*^{0} = 1

Justifying that basic identity. With that in mind:

*N*^{x-x} = *N*^{0} = 1

*N*^{x} × *N*^{-x} = 1

Dividing both sides by *N*^{x}:

*N*^{-x} = 1/*N*^{x}

With those examples, for now I’ll leave the last one for the interested reader. This second one was a bit involved — the third one is a lot easier.

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