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.

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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.

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.

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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

October 13, 2021 at 12:01am

#1019239

Meat the Press

It's a good thing I'm tackling this now. Today and going into tomorrow, I'll need to be on a liquid diet. And not my usual liquid diet, either. So I expect to be mind-destroyingly hungry, which is never how I want to be when I'm reading or writing about food.

Halfway through "Journalistic Intentions"

[18+] with

American Butchery

A while back, I encountered an article that talked about a food vendor that had been excavated in Pompeii. I don't remember what the original article was, but here's NPR's take on it.

The fascinating thing about this, for me anyway, wasn't just how well-preserved the building was; that's pretty normal for Pompeii from what I understand. Nor was it how they were able to identify nearly 2,000 year old remnants of the food served there. No, my takeaway (pun intended) from this find is how, even that long ago, people went to what are essentially fast-food restaurants. While I don't think they had drive-thru windows for chariots, the idea of walking into a place and ordering prepared food from a picture menu is not, as some would insist, an especially new, modern, or American invention.

The defining feature of civilization is specialization. And civil engineering, of course, but that's really part of specialization (they call prostitution the "oldest profession," but someone had to build a road to the whorehouse, and that someone was a civil engineer). Specialization enables a person to focus on one or a few jobs, and do them well, rather than trying to do everything themselves and fucking up most of it.

With specialization came the need to trade for services. At first, this was probably some kind of barter system, but when coinage was invented, things became much easier. Say you're a blacksmith, and you want, I don't know, wood for a tool handle. Sure, you can trade an axe head or two for some of the wood chopped, but what if the woodcutter already has all the axes he needs? Then you'd have to figure out what the woodcutter actually wants. A new pig, maybe. So then you have to make a more complex trade: horseshoes to a farmer for a pig, and then you gotta haul the pig out to the woodcutter to get the lumber you need. Coinage simplifies the process: you simply pay the woodcutter with shiny metal portraits of the Emperor or whatever, and then the woodcutter can pay for his own damn pig, leaving you more time to make swords or plowshares or whatever.

When it comes to specialization, though, few human needs are more basic than having food to eat. Hence, the butcher.

Now, I know a guy who bought a farm. Not the farm, but a farm; he's very much alive last time I checked, which was about a week ago. The idea was he and his wife could have their own livestock, and learn how to slaughter them for eatin'. Lots of people do that without necessarily owning a farm; I've also known quite a few deer hunters who were well-versed in the arcane practice of field-dressing venison.

But these two bit off more than they could chew, as they needed to have actual 21st century jobs in addition to the whole farm thing, and as I could have told them if they'd asked, trying to run a farm is a full-time job in itself (which is why I don't do it). So they took to delivering their livestock to a local butcher, who turned pigs into chops and delicious bacon. Well, it would have been delicious, but they oversalted the meat, but that's another story.

I think they made it an ethical choice: if we're going to eat meat, we should go whole hog (look, just assume every pun in this entry is intended; it's easier that way) and raise the animals ourselves, and take them through every step of the process. We're invested, that way. Of course, not eating meat wasn't an option, because meat is tasty.

Then they lost the farm, moved to a suburb, then to the middle of a city, and now they're back in the countryside again, but they get their meat from a grocery store as Nature intended.

What's my point? Well, I'm not sure that I have one, except, to get all pretentious about it, plus ça change, plus c'est la même chose. Everything sounds more pretentious in French, except maybe the word for seal (the animal with the flippers), which is phoque, and that's pronounced exactly like a certain four-letter anglo-saxonism.

Also, though, I think I'm trying to say that my friend and his wife may have had a point: it's good to know where we're ultimately getting our food. As I've mentioned too many times, I spent my childhood on a farm, so I had a pretty damn good idea already, and yet I have no problem buying ground beef at a supermarket. This guy grew up in a suburb or city or something, so didn't have that first-hand knowledge.

Some people learn the awful truth and swear off meat entirely. I don't do that myself, obviously, but I understand the ethics behind it. The steaks can get pretty high. (Okay, yeah, I forced that one.) Hell, if science ever manages to realize the potential of vat-grown meats, I'd totally switch to that. But either way, I think it's worse to mentally divorce yourself from the process of turning a steer into delicious steaks and hamburgers.

Consequently, I took the time to watch the entire video that served as the prompt for today's entry, and I'm linking it here.

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