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Category Archives: Sideband

My notes don’t include what triggered the thought, but I think it was something in one of the Lee Smolin books I read recently. My recent post, *Analog Computing*, brought the idea to mind again, because analog computers often use **op amps**. I was reminded yet again while reading about SPADs.

I’m talking about the very useful rules of thumb (heuristics) I learned to help understand, even design, electronic circuitry. They’re shortcuts in the sense of being only approximately true, but their simplified view can make a circuit much easier to understand.

I thought I’d pass them on for those interested in electronic design.

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5 Comments | tags: electrical current, electronics, transistor, voltage, volts | posted in Sideband

I’ve always had a strong curiosity about how things work. My dad used to despair how I’d take things apart but rarely put them back together. My interest was inside — in understanding the mechanism. (The irony is that I began my corporate career arc as a hardware repair technician.)

My curiosity includes a love of discovery, especially unexpected ones, and *extra* especially ones I stumble on myself. It’s one thing to be taught a neat new thing, but a rare delight to figure it out for oneself. It’s like hitting a home run (or at least a base-clearing double).

Recently, I was delighted to discover something amazing about spheres.

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9 Comments | tags: derivatives, geometry, holographic principle | posted in Math, Sideband

I’ve written a number of posts about **four-dimensional Euclidean space**, usually in the context of one of my favorite geometrical objects, the **tesseract**. I’ve also mentioned 4D Euclidean spaces as just one of many possible multi-dimensional parameter spaces. In both cases, the familiar 2D and 3D spaces generalize to additional dimensions.

This post explores a specialized 4D space that uses complex numbers along each axis of a 2D nominally Euclidean space. Each **X** & **Y** coordinate has two degrees of freedom, a *magnitude* and a *phase*. This doesn’t make 4D spaces easier to visualize, but it can offer a useful way to think about them.

It also connects back to something I wrote about in my QM-101 series.

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11 Comments | tags: 4D, complex numbers, complex plane | posted in Math, Sideband

**Trigonometry** is infamously something most normal people fear and loath. Or at least don’t understand and don’t particularly want to deal with. (In fairness, it doesn’t pop up much in regular life.) As with matrix math, trig often remains opaque even for those who do have a basic grasp of other parts of math.

Excellent and thorough tutorials exist for those interested in digging into either topic, but (as with matrix math) I thought a high-altitude flyover might be helpful in pointing out important concepts.

The irony, as it turns out, is that trig is actually pretty easy!

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30 Comments | tags: cosine, sine, sine wave, trigonometry | posted in Math, Sideband

There are many tutorials and teachers, online and off, that can teach you how to work with matrices. This post is a quick reference for the basics. **Matrix** operations are important in quantum mechanics, so I thought a Sideband might have some value.

I’ll mention the technique I use when doing **matrix multiplication** by hand. It’s a simple way of writing it out that I find helps me keep things straight. It also makes it obvious if two matrices are compatible for multiplying (not all are).

One thing to keep in mind: It’s all just adding and multiplying!

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7 Comments | tags: matrix math, matrix multiplication | posted in Math, Sideband

*Converging…*

Back in October I published two posts involving the ubiquitous **exponential function**. [see: *Circular Math* and *Fourier Geometry*] The posts were primarily about Fourier transforms, but the exponential function is a key aspect of how they work.

We write it as *e*^{x} or as *exp*(*x*) — those are equivalent forms. The latter has a formal definition that allows for the complex numbers necessary in physics. That definition is of a *series* that converges on an answer of increasing accuracy.

As a sidebar, I thought I’d illustrate that convergence. There’s an interesting non-linear aspect to it.

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9 Comments | tags: exponential function, transcendental numbers | posted in Math, Sideband

Four years ago I started pondering the **tesseract** and four-dimensional space. I first learned about them back in grade school in a science fiction short story I’d read. (A large fraction of my very early science education came from SF books.)

Greg Egan touched on tesseracts in his novel *Diaspora*, which got me thinking about them and inspired the post *Hunting Tesseracti*. That led to a general exploration of multi-dimensional spaces and rotation within those spaces, but I continued to focus on trying to truly understand the tesseract.

Today we’re going to visit the 4D space *inside* a tesseract.

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4 Comments | tags: 1D, 2D, 3D, 4D, cube, dimensions, square, tesseract | posted in Math, Sideband

The last Sideband discussed two algorithms for producing digit strings in any number base (or radix) for integer and fractional numeric values. There are some minor points I didn’t have room to explore in that post, hence this followup post. I’ll warn you now: I am going to get down in the mathematical weeds a bit.

If you had any interest in expressing numbers in different bases, or wondered how other bases do fractions, the first post covered that. This post discusses some details I want to document.

The big one concerns numeric precision and accuracy.

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3 Comments | tags: base 10, base 2, binary, binary digits, decimal, fractions, number bases | posted in Math, Sideband

*Fractional base basis.*

I suspect very few people care about expressing fractional digits in any base other than good old base ten. Truthfully, it’s likely not that many people care about expressing factional digits in good old base ten. But if you’re in the tiny handful of those with an interest in such things — and don’t already know all about it — read on.

Recently I needed to figure out how to express binary fractions of decimal numbers. For example, 3.14159 in binary. And I needed the real thing — true binary fractions — not a fake that uses integers and a virtual decimal point.

The funny thing is: I think I’ve done this before.

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4 Comments | tags: base 10, base 2, binary, binary digits, decimal, fractions, number bases | posted in Math, Sideband

Today’s earlier post got into only the beginnings of abacus operation — mainly how to add numbers. To demonstrate how they have more utility than just adding and subtracting, this Sideband tackles a multiplication problem.

This also illustrates a property of abacus operation that doesn’t arise with addition. With pen and paper, we multiply right-to-left to make carrying easier. Because of the way an abacus works, multiplication has to work left-to-right.

The process is simple enough, but has lots of steps!

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1 Comment | tags: abacus, multiplication, speed of light | posted in Sideband