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🎶

about 22 hours 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.

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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.

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

July 19, 2024 at 11:12am

#1074144

Living in DeNile

As soon as I saw this Atlas Obscura headline, I knew it was blog fodder.

Ancient Egyptians Celebrated the Feast of Drunkenness with Blood-Red Beer
One of history’s earliest mixed drinks was made to appease a ferocious goddess.

Yeah, they had me at "Feast of Drunkenness" and kept me with "Blood-Red Beer." Even though the beer would have borne little resemblance to the delicious fermented beverage we enjoy today.

There’s a whole lot of hair-splitting about who came up with the first cocktail.

If one defines "cocktail" as "the combination of an alcoholic beverage with something else," it probably happened approximately 15 seconds after fermentation was invented.

A quick Google search will reveal a litany of opinions. Some claim that the Sazerac, which first appeared around the 1830s, holds the distinction. Others would say that the unnamed concoction of spirits, sugar, and bitters—essentially, an Old-Fashioned—published by an anonymous American bartender in an 1806 newspaper is the winner.

One could also quibble that fermented beverages shouldn't count for cocktails, at which point I'd pour them a mimosa.

Last year, I spoke with cocktail historian Amanda Schuster, who has a recipe for a boozy Scottish drink with roots dating back to the 1400s in her book Signature Cocktails.

And to be fair, if I had to guess who made the first mixed drink, I'd guess Scotland.

She pointed out that people have been mixing stuff with alcohol to make it taste better for a very, very long time.

But alcohol is delicious.

According to Dora Goldsmith, an Egyptologist at Berlin’s Freie Universität, the ancient Egyptians were making mixed alcoholic beverages in the Ptolemaic Dynasty, around 300 BC.

This is where you really need to go to the link. There's a picture of her there. I wanna cast her as Morticia Addams.

While there’s evidence of distilled liquor dating back to 800 BC in parts of Asia, Egyptian “cocktails” more resembled fortified wines and beers. And when the debauched Feast of Drunkenness rolled around, these fragrant, sweet elixirs flowed freely.

You know, there are very, very few things that one can mix with beer today. Lemonade comes to mind, but not much else. Though I have had some success mixing sour beer with sparkling wine.

Goldsmith has translated what may be instructions for two such libations from a religious text. “This is a hymn, as far as the text genre goes, but behind that hymn are traces of a recipe,” Goldsmith explains.

Sounds weird to modern ears, but remember, one of the bits of evidence for the importance of beer to ancient cultures is the Sumerian Hymn to Ninkasi.

Versions of that same sacred song, known as “The Menu Song,” were chiseled onto the temples of Dendera, Philae, and Athribis.

Gotta love "The Menu Song."

“The Menu Song was probably sung on these [feast] days, while they drank this fragrant beer, and were in complete ecstasy,” Goldsmith says. “What is important to understand is that beer-brewing here is a mythological act. The brewing itself ensures that the cosmic order remains intact.”

Well, look at that, something I agree with the Ancient Egyptians on.

Before we look at the drinks themselves, let’s talk about Hathor, the goddess for whom they were created...

“She has two sides,” Goldsmith says. “On the one hand, she’s very loving, but she can also be a raging lunatic. She’s a cute little kitty cat one day and a raging lioness the next.”

Sounds like my ex.

Goldsmith attempted to recreate the elixir that got the population so wildly intoxicated by following 18 steps laid out in the hymn.

Because of course she did.

There's quite a bit more at the article (though not the recipe itself), mostly the mythological underpinnings of the celebration, which of course I find fascinating. Not because I believe them, of course, but because of what they say about the people who promulgated them.

And what they say in this case is that Egyptians loved their beer.

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