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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February 11, 2023 at 12:01am

#1044590

Carp-al Tunnel

Time for another entry in "Journalistic Intentions"

[18+].

This one, selected at random as usual, features koi: Feeding Frenzy

.

Ever notice that koi look like kaiju goldfish? There's a good reason for that, apart from most fish having a similar body plan adapted to swimming: they're both carp.

Apparently, though, they come from different subspecies. While goldfish were bred in China over a thousand years ago,

selective breeding of koi is a more recent Japanese development. For the benefit of my fellow Americans with a limited sense of historical time, by "recent," I mean about 200 years ago.

And goldfish aren't really gold. By which I mean, no, of course they're not made of the chemical element Au, but their coloration always struck me as being more orange than yellow, though it can vary. Not everyone calls them goldfish, either; in French, the name is poisson rouge, which it doesn't take three and a half years of French lessons to know means "red fish."

They're not red, either, but different cultures have different conceptions of color.

No well-known metals are red or orange, so "goldfish" it is.

Koi obviously aren't limited to the orange color, but the point here is that none of these fish are the product of natural selection, unless you take my philosophical stance that we're part of nature, and so anything we accomplish, from air pollution to plastic to miniaturized orange carp, is also natural. "Artificial selection" or "selective breeding" are perfectly good terms for the process, though.

This strikes me as similar to dog or cat coloration. I don't mean different dog breeds, but, like, how a collie can be sable and white or black, sable and white. Or the different patterns of spots on dalmatians. While some koi resemble calico cats in coloration, the calico trait apparently doesn't breed true in cats, being instead a genetic thing associated with the feline X chromosome.

I have heard, but can't confirm, that before we nearly extincted tigers, some of them could also display the calico trait. Which I'd pay to see. As tigers are largely Asian in range, well, that brings us back to the origins of goldfish and koi.

All carp, incidentally, are food. I mean, everything is edible once, but carp has been a staple in many cultures, despite difficulties in its preparation (it's bony as fuck). In Eastern Europe, it can become gefilte fish, which, well, some of my people consider that to be food, whereas I do not.

As pretty as they are in koi ponds, the Asian carp is considered an invasive species in the US. I wrote about an attempt to contain them on the Mississippi River in a blog entry a while back: "Dam It"

. In that entry, I said:

That lock was closed a few years ago, but the structure remains. The reason for the closure was to stop the spread of Asian carp, an invasive species, upriver. So when I was touring the lock, looking into the waters upriver, what do you think I saw? Go ahead, guess.

That's right. An Asian carp.

On the plus side, I guess I caught my first Pokemon.

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