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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Write_Mikey_Write!

about 6 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

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Adherennium Plotting something

about 8 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

January 10, 2022 at 12:01am

#1024433

I Don't Have The Guts

From "JAFBG"

[XGC], a discussion of intestines.

Tell us about a time when you had a gut instinct about someone or some situation that turned out to be 100% accurate.

I find it interesting to examine some of the metaphors we use for certain feelings. Several emotions including love are, as we all know, attributed to the heart. Guts are the source of fortitude or courage (when we're not using gonads as the metaphor), in addition to the sense used in today's prompt. Sometimes when you're absolutely certain of something, you "feel it in your bones." When it's hard to accept something, we sometimes say we can't stomach it. And, while it's not a body part but rather created by body parts, bile is traditionally associated with strong emotions such as anger.

In consensus reality, though, all these things are products of consciousness, which is largely a brain thing. Still, the metaphors persist.

I'm not entirely sure, but I think this all has to do with classical attempts at medicine, at least in the West. You know, the whole "balancing the humours" thing or whatever. Similar systems exist in Eastern traditions; the one I'm most familiar with involves chakras

from Indian culture, which attribute different aspects of being to certain centers in the body.

All of this is not to provide a crash course in ancient spiritual traditions, which I'm not qualified to do anyway, but to point out that it seems to be a nearly universal human trait to associate different parts of the physical body with different aspects of the mind -- even if the exact associations might be different between cultures.

But what I find the most interesting about all this is that, more recently, Western scientific, evidence-based medicine has indeed found links between mind and body; in fact, one could say that drawing a solid line between "mind" and "body" might not be as easy as was once thought.

For instance, we know that stress affects the body as well as the mind. And that gut health -- the presence and function of certain species of microorganisms that exist in our digestive tracts -- is somehow connected to mental health. Which direction that correlation goes, I can't say, but from what I understand, people with messed-up intestinal flora tend to have mental issues as well.

Incidentally, it appears that the appendix

, long ignored as a vestigial organ whose only function appeared to be its ability to get infected and kill you if they don't surgically extract it, actually might serve a purpose: as a reservoir for gut bacteria that can replenish the intestinal tract if, through illness or poisoning or whatever, it loses its natural ecosystem.

Wish they'd figured that out before they yanked my appendix, but whatever.

But I digress. The gut biome

is a thing, though, and they're still figuring out stuff about it. The interesting thing is that these microorganisms are their own species, and the number of them exceeds the number of human cells in our bodies. (As I understand it, this is possible because they are much smaller than animal cells.) And they do not share our DNA. (That might not be an entirely accurate statement because of arcane concepts like horizontal gene transfer and other stuff I don't fully understand, but on a very simplistic level, that's my understanding.) But without these non-human entities which are nevertheless a part of us, we wouldn't function properly, if at all.

All of this is a roundabout way of saying that the "gut instinct" referred to in the prompt might well be the product of biochemical communication with your intestinal flora, which, as living microorganisms, have their own evolutionary drive to survive.

Or maybe not. This ventures into the realm of science fiction; I don't have any actual data to back it up, so just consider that to be speculation, but one that returns us to the older realm of body-part metaphors for emotion -- except that maybe it's not a metaphor, after all.

In any case, I'm saying all this in order to avoid actually addressing the prompt -- because I couldn't think of a single situation where I had such a feeling that definitively turned out to be right. It's like... maybe you see a shifty-looking person hanging out in a doorway at night, and so cross the street to avoid them. Would they have mugged you? Maybe. Maybe not. But that just sounds like ordinary prudence to me. The only way to know if your "gut instinct" was right would be to watch them and see if they mug someone else, and why would you do that instead of just getting to a safer place as soon as possible?

Sure, sometimes I've done something because I felt like it, and it turned out to be the better course of action. Like if I can take two roads, and I just suddenly decide to take the scenic route, and I find out later that the faster route had a massive traffic jam. Was it a feeling? Fate? More likely it was just the luck of the draw, and it could easily have gone the other way around. You tend to remember your lucky random choices and dismiss the unlucky ones.

In any case, at this point my microorganisms are requesting nutrients, and who am I to question my unicellular overlords?

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