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Category Archives: 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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28 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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3 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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8 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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3 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

This is a Sideband to the previous post, **The 4th Dimension**. It’s for those who want to know more about the rotation discussed in that post, specifically with regard to axes *involved with* rotation versus axes *about which* rotation occurs.

The latter, rotation about (or around) an axis, is what we usually mean when we refer to a *rotation axis*. A key characteristic of such an axis is that coordinate values on that axis *don’t change* during rotation. Rotating about (or on or around) the **Y** axis means that the **Y** coordinate values never change.

In contrast, an axis *involved with* rotation changes its associated coordinate values according to the angle of rotation. The difference is starkly apparent when we look at rotation matrices.

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3 Comments | tags: 2D, 3D, 4D, column vector, matrix math, matrix transform, rotation, rotation matrix, unit vector, vectors | posted in Math, Sideband

In the last installment I introduced the idea of a *transformation matrix* — a square matrix that we view as a set of (vertically written) vectors describing a new *basis* for a transformed space. Points in the original space have the same relationship to the original basis as points in the transformed space have to the transformed basis.

When we left off, I had just introduced the idea of a *rotation matrix*. Two immediate questions were: How do we create a rotation matrix, and how do we use it. (By extension, how do we create and use *any* matrix?)

This is where our story resumes…

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3 Comments | tags: 3Blue1Brown, column vector, linear algebra, matrix math, matrix multiplication, rotation, unit vector, vectors | posted in Math, Sideband

For me, the star attraction of March Mathness is **matrix rotation**. It’s a new toy (um, *tool*) for me that’s exciting on two levels: Firstly, it answers key questions I’ve had about rotation, especially with regard to **4D** (let alone **3D** or easy peasy **2D**). Secondly, I’ve never had a handle on matrix math, and thanks to an extraordinary YouTube channel, now I see it in a whole new light.

Literally (and I do mean “literally” literally), I will never look at a matrix the same way again. Knowing how to look at them changes everything. That they turned out to be exactly what I needed to understand rotation makes the whole thing kinda wondrous.

I’m going to try to provide an overview of what I learned and then point to a great set of YouTube videos if you want to learn, too. Continue reading

18 Comments | tags: 3Blue1Brown, column vector, complex numbers, linear algebra, matrix math, matrix multiplication, rotation, trigonometry, unit vector, vectors | posted in Math, Sideband