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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Brandiwyn🎶

a day ago
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.

3 Like

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.

1 Like

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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

December 12, 2020 at 12:01am

#1000093

Red and Redder

"30-Day Blogging Challenge ON HIATUS" [13+]:
12. Pointsettia Day
According to Mexican folklore, there is a story of a little poor girl who had nothing to bring to church for Christmas. On her way to church, she picked some plants by the side of the road. As she entered the church, the leaves at the tips of the branches turned into bright, brilliant red flowers. You guessed it... Poinsettias.
Write anything you like about Poinsettias.

"JAFBG" [XGC]:
They say blood is thicker than water. How does your family prove this saying wrong?

I think one of the most interesting subgenres of mythology is what Kipling popularized with the "Just So Story," making shit up about how things got to be the way they are. Apparently the technical term is "etiological myth," which I found on Wikipedia. Etiology itself is the study of origins, and one can approach it mythologically or scientifically.

My favorite etiological myth is from the Lenape and involves the turkey vulture (big surprise there) and several other animals. This is the version I could find.

Now look, I'll be the first to admit that I rarely click on links others provide; I just can't be arsed. So maybe I'm being hypocritical to expect you to click on the above link. But I'm asking you to do it, anyway, because I don't want to just copy all the text into here and make this post even longer and more difficult to get through. It is one reason I consider the turkey vulture to be the closest thing I have to a spirit animal.

It is, of course, fiction, a made-up story. Such folklore says nothing at all about how things got to be the way they are in reality, but it speaks absolute volumes about how we humans look at the world -- first by coming up with such legends, and second by how they are passed down.

Science, of course, approaches etiology in a more systematic way. The scientific explanation for the vulture's featherless head has to do with its adaptation to its niche in the ecosystem, not the vulture's ancient self-sacrifice for the good of all. Both stories are fascinating, each to its own purpose. The important thing is to be able to separate reality from fantasy.

Like with the poinsettia. Admittedly, I didn't look very hard for the evolutionary explanation for the red bits -- probably something to do with warding off predators, or maybe attracting certain pollinators; those are the usual "reasons" for any plant coloration that isn't green. But the reason I quit looking is because I found another, more modern myth about the poinsettia.

Honestly, I never paid those plants much attention. To me it was a Christmas thing and Christmas things ward me off like garlic does vampires. But one thing that I heard repeated, over and over: "Watch out for poinsettias if you have children or pets. They're highly poisonous!"

This turns out not to be the case.

Yes, I just included yet another link for you to click on. Summary: Poinsettias aren't edible, but they also aren't toxic.

And yet, this is the first I've heard of that, because the myth is so persistent. Unlike the myth about the little girl going to church, though, people don't know it's a myth, and they repeat it as pure truth. Which, again, it is not. Now, is it a harmful falsehood? Probably not; it's usually better to assume that something is dangerous than to assume it's perfectly safe. And yet it is a lie, a falsehood, a thing with little to no basis in fact.

So it is (SEGUÉ) with "Blood is thicker than water."

A proverb isn't a plant. It has no existence or reality beyond people repeating it; while poinsettias would still be there if humans were to suddenly disappear with all of their records, that phrase would vanish as if it never was. So it's even more a reflection of human ideas than creation myths are.

I've heard it said that the "original" expression was "the blood of the covenant is thicker than the water of the womb," but as with poinsettias' toxicity, that appears not to be the case. No, as far as I can tell, the original meaning was indeed that genetic family ties are stronger than others.

But that's the kind of thing that's only true if you think it's true. And that can be a dangerous fiction. For instance, presumably if you get married, it's to someone who isn't too close a relation, right? Like, I know we're all related to some extent, but at least you don't go with a sibling or first cousin or whatever. So you take an oath, a vow, to support each other. But if you believe in the "blood is thicker than water" myth, then your responsibility to your sibling or parents would supersede responsibility to spouse.

Maybe you do feel that way; I can't say that it's wrong, exactly. But to me, an oath that you enter into willingly is far more binding than one that is a mere accident of relation. Like when my parents adopted me. No close genetic relation there, but they fulfilled all the necessary responsibilities and then some (me, not so much, probably, but then, it wasn't an oath *I* took). Blood? Water? That's all metaphor anyway. What matters is commitment.

And on the other side, if your family is toxic, I don't feel like you should continue to put up with them because of some mythology about "blood" bonds.

Like I said, it's important to discern reality from fantasy, and while that's easier in some cases than others, one shouldn't believe every little saying that comes down the road. Poinsettias aren't poisonous. Family sometimes is.

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