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.

2 Like

April 17, 2023 at 9:16am

#1048319

In-Vesting

Time for another clothing-related prompt from "Journalistic Intentions"

[18+]. Unlike earlier prompts, this time, the RNG gave me something I've actually heard of.

Waistcoat

Now, here's a refreshing change: a name for an article of clothing that actually makes linguistic sense. Sort of. Except that "waistcoat" and "vest" are synonyms. And also, some sources claim the name is a pun: the vest was made from the waste material from tailoring the jacket and pants, hence, waste-coat. As it hugs the waist (not to be confused with a cummerbund, which is worn around the waist), the pun was really inevitable.

I've long been operating under the assumption that it's a "vest" when it's the outer layer of an outfit, and "waistcoat" when it's worn as, say, part of a three-piece suit. Three-piece suit is, itself, a misleading phrase, as one almost always also wears socks, shoes, a shirt, and a tie, making it, at the very least, a seven-piece suit. Not to mention underwear (bottom and top) (optional), cufflinks, a belt or suspenders, a hat (some of us still wear them), and my favorite, the pocket square. Though I'll grant that you normally buy those accessories separately, with the trousers-waistcoat-jacket part sold as a unit.

That assumption turns out not to be the case.

Of course, it's possible to have trousers, waistcoat, and jacket be of different materials, colors, and patterns. But then it's not a three-piece suit, is it?

Look, I know more about traditionally men's fashion than women's. This should not be a surprise to anyone.

As an aside, I mentioned up there that "vest" is effectively a synonym for "waistcoat." This isn't strictly true. In the UK, a "vest" is what we Yanks unfortunately have taken to calling a "wife-beater," from the stereotype of the man wearing only a sleeveless undershirt who engages in domestic abuse. This is, of course, not the only fashion-related linguistic trap between US and UK English. "I'm wearing pants" has a whole different meaning, too.

Adding to linguistic confusion is that "to vest" is a verb that has several meanings, including "to put on clothes." Whether those clothes involve waistcoats or not.

All of which is to say that even in this relatively simple case, nothing in fashion is as straightforward as one might think.

For example. I've been wearing suits (occasionally) (yes, I own suits) since I was a kid, and it took me until like 2019 to discover that you're not supposed to close the bottom button of a waistcoat. I knew about that rule with jackets, but no one told me about vests.

As an engineer, this really annoys me beyond all reason. While I understand, intellectually, the idea of nonfunctional junk on clothing (the rivets on a pair of Levi's served a purpose once, but nowadays they're just tradition, for example), why make it look like you're supposed to button something when you're not?

The only answer I can come up with is that they do it as a visual kind of shibboleth: if you don't conform, then those in the know have yet another reason to look down upon you.

Which is why I love pocket squares. They make excellent gags.

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