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.

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.

1 Like

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

October 20, 2022 at 12:02am

#1039439

Against the Grain

This is just something I thought was interesting.

This Nearly Lost Ancient Grain Tradition Could Be the Future of Farming
A past global staple you’ve never heard of, maslins are poised for a comeback.

And yes, I'd never heard of maslins.

And instead of sowing the seeds of a single grain in orderly rows, spread a mix of grains all over the field, “mimicking nature so crops have random distribution patterns, as in natural forests,” he says. Once harvested, these grain mixtures could be turned into many things: nutritious bread, a kind of roasted-grain trail mix called kolo, beer, and the potent clear spirit known as areki.

You had me at beer.

“We’ll plant the things that go together and are compatible with each other,” Zemede says. “Our farmers are good at mirroring nature.”

One might ask, if you're just going to mirror nature, why not just take what's already in nature? Well, there's just not enough of it for our teeming masses.

Ethiopia is one of the few places in the world where farmers still grow maslins, the general term for different varieties and species of grain that are sown in the same field, or intercropped. Maslins sustained humans for millennia, possibly predating the rise of agriculture more than 10,000 years ago.

Predating or predicting? Or was it a step in that direction?

These grain mixtures tend to be more resilient to pests and drought, and to lend more complex flavors to breads, beer, and booze.

And now I want to try some.

Worldwide, maslins fell out of favor long ago, replaced nearly everywhere by sprawling, single-grain monoculture—but a small and passionate group of scientists, including Zemede, is hoping to change that.

There are, of course, advantages to monocultures (the article goes into that a little bit), which is why people plant single crops in the first place.

The fact that maslins are grown today only in Ethiopia and pockets of Georgia, Eritrea, and a handful of other countries belies how widespread they once were.

One of those countries is not like the others...

One time, I was in Vegas and found out a dealer was from Eritrea. "Do you know where that is?" she asked, clearly expecting ignorance.

"East Africa, near Ethiopia," I said.

She was, indeed, surprised. Look, I don't know everything, but I'm pretty good at looking at maps.

Today, Ethiopian farmers are feeling the pressure to grow modern monoculture crops, thanks in part to a national push to become an agricultural powerhouse. “If you export grains, you want them to be uniform,” says McAlvay. “The global market wants a certain type of wheat for their Wonder Bread. A mixture of three varieties of wheat and four varieties of barley with some other things thrown in really doesn’t make the cut.”

The obvious problem there is that Wonder Bread is basically shit.

The grain mixes also appear to have natural resistance to pests, from insects to fungal diseases. While a pest adapted to attack one species of grain will have a field day, no pun intended, when set loose in a monoculture crop, it won’t be able to jump from plant to plant if the individual it attacks is surrounded by other kinds of grain, McAlvay explains.

No. Own it. Own that pun. Revel in it.

There's a lot more at the article, which, like I said, is just something I found interesting. I just hope I don't have to go all the way to Ethiopia to try the beer.

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