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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 11 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

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.

6 Like • Comment • View 4 comments

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