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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Write_Mikey_Write!

about an hour ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 3 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

November 1, 2023 at 10:07am

#1058428

Shorter Hops

Well, BBC, if you're trying to frighten me into once again giving a shit about climate change... congratulations.

Climate change could make beer taste worse

Imagine a clutter of hipsters squatting around a crackling campfire. Shadows flicker, unexplained lights glimmer from the inky depths of the surrounding forest. Engaged in the time-honored tradition of telling scary stories in the darkness, one of them holds a flashlight under his bearded face, casting unnatural shadows and highlighting where he'd forgotten to snip off a stray whisker strand.

He pauses in his tale, scanning the rapt audience until he's sure he has their undivided attention. "And then," he intones, "they discovered that all of their beer... had turned into Coors Light!"

Everyone: "Gasp!"

Yeah, they're hipsters, so they actually say "gasp" instead of gasping.

Anyway. I don't think this article is saying it'll be quite that bad, but it's scary enough:

Global warming is changing the quality and taste of beer, scientists have warned.

"Weather events will get more extreme." "Meh."

"Sea level will rise, inundating coastal cities." "Whatever; the people can move."

"We'll lose agricultural production." "So? It'll just relocate."

"Beer will taste worse." "Holy fuck, something needs to be done right now!"

Though I don't know if it'll really change any minds. We already know it's going to destroy chocolate and coffee production, and still not enough people care to do anything about it.

Hotter, longer and drier summers are predicted to worsen the situation, and could lead to beer becoming more expensive.

Beer always becomes more expensive. If it ever drops in price, they tax it to make up the difference.

Hops, the flower of the hop plant, are the crucial fourth ingredient in the beer brewing process - alongside water, yeast and malt. They are added before the boiling process to add bitterness, but can also be added afterwards to change the overall flavour.

Yeah, thanks, but no. I mean, yes, adding hops is important for flavor and aroma, but the reason we started with hops in the first place is that they're a preservative, and keep the beer from going bad as quickly. Especially important in the time before refrigeration, or after we stop producing electricity.

Farmers have been working to adapt their growing practices to improve yields, such as moving farms higher up valleys where there is more rainfall, and installing irrigation systems.

Irrigation systems are great, until their sources dry up, too. Which they will.

In summary, we're doomed, and we won't even have decent beer to drown our doominess in.

5 Like • Comment • View 4 comments

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