Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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December 11, 2022 at 12:01am

#1041676

Revisited: "Deep Dive"

When I pull the levers on the Wayback Machine, I exclude any entries from the last 12 months. Today's revisit is, however, from last year; specifically, July, after my cross-country trip was cut tragically short by a deer. Well, tragically for the deer; for me it was part minor inconvenience, part disappointment, and part unexpected adventure.

"Deep Dive"

This was a response to a then-current prompt from "JAFBG"

[XGC], one of the blogging activities around here that I sometimes do. It encourages rants, and I like to rant, so, big wins all around.

The entry my random number generator landed on was not, however, particularly ranty, just some discussion from a prompt about comfort zones. It's also recent enough that I don't think I've changed much as a person since then, and I mostly remember the context. But I still have some things to expand on.

Every time I travel, I step outside my comfort zone (which is about a three-mile radius centered on my house).

Oh, sure, there are times I can step back in, even when traveling. Most of the time, I feel at home in a brewery or brewpub.

Yesterday, I went to a beer festival. I always loved beer festivals, where a bunch of different brewers set up tables with, usually, 2-4 taps of their product. There weren't any of those for over 2 years. This particular one was kinda expensive, and only had ten vendors, including one vineyard, one cidery, and one sake brewery. Yes, we have a sake brewery here, one of the few in the US, and it's a good one.

The one advantage of having so few producers represented is that I was able to try one of everything. Of course, by the time I was done, I was pretty lit. I barely remember the Uber trip home. Point being, I was well within my comfort zone.

My near-visioin is still pretty clear, so often, what I'll do is take a picture of the taps or the bottles with my phone, and hold the phone up close to my face to read the words.

I find it somewhat amusing that I typo'd "vision" in that sentence. At the time, I was using prescription reading glasses to type, ones that also accounted for the cataracts I had at the time. They weren't perfect. Pretty sure that's not the only typo I made in that period. I don't think I misspell things very often, so when I do, it's somewhat embarrassing.

Speaking of comfort zones, I've lived most of my life with near-perfect vision, with only a slight astigmatism to deal with -- this was only a real impediment to stargazing, and I never even noticed it in my daily life. Later in life, I learned to deal with needing reading glasses for computer work and books. Now, in the space of two years, I've gone from near-perfect distance vision to not being able to read beer taps from across the bar. That's uncomfortable, and I don't like it.

I don't remember if I noted this at the time: when I went in for a cataract surgery consultation last year, the ophthalmologist went over some different options for replacement lenses. The cheap ones, the ones insurance covered, would be the kind that only worked for distance vision; I'd need glasses for anything closer. There was also an option not covered by insurance that worked for both near and far. But, as he explained, the side effect would be a halo/starburst effect around light sources at night. "Usually that's only a problem if your hobby is astronomy," he said.

Well, you know me. Astronomy is one of my hobbies.

Still, balancing that with its benefits, I opted (pun intended) for the fancy ones. The surgery was almost a year ago, now. Indeed, lights (and stars and planets and the moon) have a halo/starburst effect. Worth it for not needing glasses. As it was in my younger days, I can switch from reading a screen to watching a far-away cloud, both with roughly equal clarity.

Sorcery.

My vision with the upscale lenses isn't perfect, but it's good enough, and definitely better than having cataracts.

So, would I dive into a dive again? Sure. But I think the real question is, "Would you step out of your comfort zone again?" And the answer to that is still yes.

At least, yes, once I buy a car again, which I might just put off until after cataract surgery. Which itself is so far out of my comfort zone that it might as well be on a planet orbiting Proxima Centauri, and an experience that, once I have it, I doubt I'll be in any hurry to repeat.

So, as I noted in here, I did indeed buy another car—though not until this past October. Not because of vision, but because availability was low and prices were high.

I feel another road trip coming on, too. Difficulty: possibility of winter weather. We'll see. It'd be another affront to my comfort zone, but at least the car is comfortable.

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