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.

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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.

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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.

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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.

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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.

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October 13, 2023 at 7:44am

#1057294

Amethyst

Amethyst
an entry for "Journalistic Intentions"

[18+]

Way back in the murky mists of deep time, during a period when I was on the fence about being childfree or not, I knew what I wanted to name a daughter: Amethyst. That would even be my red flag, I decided. When I'm dating someone, I find out if she likes the idea of naming a girl Amethyst and, if not, we weren't going to go any further.

This lasted, oh, about a month or so, when I dated a woman who hated the idea, but was really smoking hot, so all of those plans went right out the window. It wasn't long after that, probably, that I decided my actual red flag was "I want kids."

But I digress; the point is, I liked the sound of that word for as long as I can remember. Which is actually a fairly long time, as I was told early on that it's my "birth stone," just because I arrived in February.

"Birth stone" is, of course, a transparent marketing gimmick, like those silly lists of anniversary gifts. Regardless, I liked the sound of the word amethyst, and I liked the deep purple tint of the stone.

It's just quartz, you know. Silicon dioxide, the second most common mineral of the Earth's crust. (The first is feldspar, which really shouldn't count, as it isn't always composed of the same elements the way SiO₂ is.)

Of course, amethyst isn't really "just" quartz. That color comes from the occasional iron atom in the crystal lattice. Don't ask me why that makes it purple, though; it probably involves quantum effects.

So yeah, common or not, I have a thing for amethyst. Or, at least, I did, until I found out the etymology of the word. It's probably common knowledge by now, but I'll reiterate it here anyway: the word comes to us from ancient Greece, though they certainly weren't the first people to know about the mineral. They assigned it the mystic property of protecting a person against drunkenness, so they named it not-intoxicated, or, in their words, a-methyst.

From what I understand, they even made goblets of carved amethyst on the theory that you could drink all you wanted to out of them and not get drunk. If they'd been half the scientists people think they were, though (and I have an article in the queue that touches on the ancient Greek penchant for natural philosophy), they might have done controlled, double-blind tests and realized that no, it possesses no such property, and any perceived resistance to the blessings of Dionysus was essentially a placebo effect.

I don't know how this belief didn't piss off Dionysus. And you don't want to piss off Dionysus; he's a mean drunk. I, however, am not a mean drunk; I'm a lot meaner when I'm sober. So even though didn't possess this magical quality, the mythology made it lose some of its sparkliness for me.

Therefore, it's just as well I never had kids to saddle one with a name I'd grow to distrust.

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