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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April 24, 2023 at 9:50am

#1048696

It's In My Genes

It is now time for me to wrap up my April entries into "Journalistic Intentions"

[18+]. Of the nine prompts I haven't tackled yet, which one will my random number generator choose this morning? Let's find out...

denim

Oh, glorious.

My dad always called blue jeans "dungarees." His attire of choice was denim overalls, because by the time I became self-aware, he had retired from the military and become a farmer, and just as military officers wear uniforms, farmers wear overalls. He didn't call the overalls "dungarees," though. I don't know why and I certainly can't ask him now. Probably, like about 80% of what he did, it was to annoy me, because I liked wearing blue jeans and hated the word "dungarees" because it started with the sound "dung," which, as farmers, we were all too familiar with. Still, many people, especially in his day, used the words interchangeably.

Nowadays I like wearing black jeans, often faded to what my friends called "gamer gray," but while the color and the waist measurement have changed, they still come from Levi Strauss, complete with rivets at the seam junctions.

Turns out that denim

and dungaree

fabric, while similar, aren't actually the same thing. The latter was apparently invented in India many centuries ago, but the particular combination of dye and weave that go into a pair of blue jeans dates back to maybe the 19th century, with the invention of blue jeans coming shortly after the end of the US Civil War.

If you bother going to those links above, you'll see what people like to ding Wikipedia on: one calls dungaree a predecessor or precursor to denim, while the other asserts that the connection is unclear. This is why consulting footnotes and other sources is important. But I'm aware that's a lot of work, and in spite of my predilection for wearing what was originally a working-class pair of pants, I'm allergic to work.

No disrespect intended to the Indian subcontinent origin of dungaree—that culture invented many essential things for society, including our counting numbers (they're called Arabic numerals, but it turns out the Arabs borrowed the system from India). It's clear to me, though, that denim as we know it today had its origin in the same place so much fashion came from: France. Specifically, the town called Nimes, in the south. De Nimes apparently got Anglicized as denim.

Which makes me think that "jeans" should be pronounced "zhahns" in the manner of Jean-Luc Picard, but that turns out not to be the case. Apparently (at least according to one of the non-Wikipedia sources I looked at), its etymology comes from the port of Genoa, which in French is or was called Genes, not to be confused with chromosomes. Why this is the case, I couldn't be arsed to figure out; I did look at a map, which shows that while Nimes is close to la mer Méditerannée, there are at least two perfectly good French ports (Marseille and Nice) between Nimes and Genes.

In any case, you know what the French word for jeans is? It's not pronounced like Admiral Picard's name. No, it's pronounced almost exactly like Anglophones pronounce it (the 'j' sound is a bit different), which must be a source of early heart attacks for members of l'Academie française.

But jeans as we know them aren't French; they're indisputably a Jewish invention.

Gotcha. No, Levi Strauss (German-Jewish family) didn't invent blue jeans; a tailor of Lithuanian-Jewish origin named Jacob Davis (last name changed upon immigration, as with many immigrants) invented them (or at least perfected their durability by using rivets) in Nevada (which had just barely become a US State), but apparently couldn't keep up with demand, so he partnered up with Levi-Strauss to meet production needs.

The rest, as they say, is history.

To summarize, it's an Indian inspiration for a French fabric that was perfected as work clothing by Jewish immigrants in the Wild West of the US. It would be difficult to get more multicultural than that, but I suspect you can find other examples of intercultural fashion development if you try.

I mentioned in a previous entry the persistence of rivets in Levis despite advances in clothing technology that render them mostly decorative now. At the time of their invention, though, riveting the weak spots was the innovation that made jeans the preferred outfit of the working class.

They never did, as far as I know, rivet the crotch. I mean, I can kind of understand why; metal tends to get hot in the sun, and the last place you want to heat up is right under your balls. Or, if you don't have them, your cooch.

So that's what always fails first for me on a pair of black Levis (though usually only after a few good years of stalwart service). Hopefully not while I'm in public. Nothing's perfect.

But in a world dominated by throwaway fashion, it's good to know some clothing is still built to to stand the test of time.

Speaking of which, there's one final thing I want to add: that little pocket on the right where you might keep an emergency quarter or two, or a key? The original purpose of that was to hold a pocket watch. Because no matter how hard you work, you still need to keep track of time.

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