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

October 27, 2023 at 10:14am

#1058145

Carmine

Carmine
an entry for "Journalistic Intentions"

[18+]

I always wondered about "carmine." Is it a bomb you put on a car to make it blow up? A hole in the ground where you dig for cars? Or is it pidgin for "my car?" Oooh, I know: it's a corruption of "karma mine," which is a place where you take out only what you put in.

Well, no. Sadly, it's none of those things, although they would absolutely make the world a more interesting place.

It is a not-unpopular name, especially for those of Italian origin, and one with several cognates, including Carmen (as in Sandiego) and Carmelita. But it turns out that the name has two origins, like a river with a fork: one from Latin, meaning song, and one from Hebrew, meaning garden. Not that famous "first" garden, though; that was something like paradise. (No, really, the original Hebrew word for it is also translated as paradise.) Just your regular old garden, though presumably one with flowers and bushes rather than cabbages. If one were poetically inclined, one could say that the name Carmine might mean "garden of song," which would presumably be attached to Leonard Cohen's "Tower of Song."

The way Hebrew might have gotten mixed up with Latin to create that name, I leave as an exercise for the reader.

I say all this because one might be forgiven for thinking that the name and the word for a certain deep red color share an origin. They do not. Though one Carmine I heard of, Carmine Infantino, was a comic book artist well-known in some circles (comic readers) for a great run on the Deadman series, and Deadman wore a carmine-colored onesie. Coincidence? Well, actually, yeah. Infantino was drawing an established character. Also, only the reader (and maybe other dead people in the comic) could see Deadman, as he was a ghost. His superpower was possession, incidentally.

What's not a coincidence is that carmine and crimson, the color words, share a linguistic root. In the analogy above, that would be two rivers sharing a single source, which is kind of uncommon for rivers when compared to the other way around, but not so rare in language.

No, carmine-the-color came from bugs.

Apparently, there was this one species of insect which, when tortured, killed, and ground up, produced the carmine dye. I have no idea if this is still the practice, but at least no insects were harmed in making that color in your photo editing software. After all, something's gotta live in your garden.

2 Like • Comment

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