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.

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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

October 20, 2023 at 11:11am

#1057739

Navy

Navy
an entry for "Journalistic Intentions"

[18+]

Of all the weird, strange, or just plain incomprehensible names for colors, "navy" stands out as one that almost makes actual sense... from a certain point of view. Under certain ideal conditions, the ocean appears that deep, dark blue color, for reasons I can't be arsed to go into right now but you're right; it does involve physics.

But our word, navy, comes from a Latin word that referred more to ships than the ocean. Still, you know, you can't have a ship without something to float it in. Which reminds me that another word from the Latin navis is navigate, whose meaning should be limited to finding one's way around on a body of water. But, naturally, it's not, because we're pretty good at making new meanings for old words, and new words for old meanings.

Take, for example, space. Thanks to years and years of science fiction, we know what a vessel that carries things around in space is: a spaceship. And again, that makes sense from a certain point of view. But the harsh reality is that if we do get to the point where we have spacecraft transporting people or things around out there, the vessels will have far more in common with a submarine than they do with a ship. And, by naval convention, submarines are always boats, not ships.

"Spaceboat" just doesn't have the same ring to it, though, does it?

Don't ask me to define the difference between boats and ships further. My dad was a sailor, and I never fully understood the distinction he made, when he bothered to make one. Near as I can tell, a boat goes out from port or ship and returns to the same port or ship, while a ship carries cargo and/or passengers from one port to another (hence the verb "to ship," which also refers to sending parcels by road or rail). But by that definition, a ferryboat should be a ship, but it's not (I sidestep this by calling it a "ferry"). A ship can carry a boat, but a boat can't carry a ship, though a tugboat can push a ship, despite "tug" having the connotation of "pull."

Anyway. The other thing we get from science fiction is the use, in spaceships, of a "navigator," like on the original Enterprise. Properly, this should be "astrogator," but that would just lead to alligator puns (though some SF does use this term), so probably best to repurpose the word. And don't get me started on the etymology of "bridge," as in a spaceship's control room, which takes its origin from Mississippi River steamboats... dammit, I said don't get me started.

Still, sometimes it bugs me that landlubber GPS uses "navigation." I mean, we want fewer people not paying attention and driving their cars into lakes, right? But again, there's not much to choose from in terms of better words. "Orienteering" is the process of finding one's way around on land, but that doesn't really work in cars or trucks, does it? So we're stuck with navigation.

But even that makes more sense than calling a web browser a "navigator." Which you don't see much these days in English, but one early web browser was Netscape Navigator. And the French word for browser is "navigateur." This wouldn't bother me so much if the defining metaphor of the Internet weren't a spiderweb, rather than an ocean.

You could say all the contradictions give me the blues.

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