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 13 hours 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 15 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.

April 13, 2022 at 12:02am

#1030623

Zoned Out

An entry for "Journalistic Intentions"

[18+]...

Gulf of Mexico Dead Zone

From what I've heard, the Earth looked a little different 65 million years ago.

Here's a drawing: https://images.slideplayer.com/16/5101171/slides/slide_2.jpg

Some of the continents are vaguely recognizable. You can make out South America, Africa, Australia as it was divorcing from Antarctica. In between Africa and Australia, just east of what is already recognizable as Madagascar? That'll become India in several million years as it migrates north and forms the Himalayas.

And that big explosion-looking thing in red? In what's going to become the Gulf of Mexico? That's the impact crater that they say probably killed off most of the dinosaurs.

Though they're still speculating about that, last I heard, it remains the most likely scenario. Whatever the actual cause, there's little doubt that it was around that time when the non-avian dinosaurs died off, paving the way for mammals to take over, which eventually led to the evolution of some clever apes who, in the span of about 100 years, wiped out a lot of the deposits of crude oil that it took billions of years for nature to sequester underground.

What is certain is that the impact would have killed off anything in its immediate vicinity. The rest was, so the speculation goes, devastated by the resulting sudden change in climate.

It's a myth that crude oil is dead dinosaurs. A myth, but a persistent one. Sinclair still uses a dinosaur for its logo, which was originally a brontosaurus, then an apatosaurus, and now they're saying no, wait, it's actually a brontosaurus after all.

No, the bulk of these deposits were formed long before the dinosaurs. Had they developed oil wells, perhaps they would have sucked the planet dry before we even showed up on the stage. But they didn't, so it waited another 65 million years or so until we came along and decided we needed energy.

And one of the places we get that energy from happens to be the Gulf of Mexico. We all know about the massive oil spill there from over a decade ago, but that wasn't the only one. These spills killed a bunch of shit and drove the rest off.

Another Gulf of Mexico dead zone. Not from a sudden impact, but in geological time, 100 years might as well be a minute.

People can't help but speculate: what comes after humans, after we've experienced the inevitable extinction event? Will another civilization arise? Maybe cockroaches will learn to build and drive cars?

Well, no. Because we've sucked all the oil out that powered those cars. And taken away all the low-hanging fruit -- the once-easy-to-find minerals that kick-started each of what we like to call "ages:" stone, bronze, iron, nuclear, whatever. It will take millions, even billions, of years to recover those, if it happens at all, as continents continue to drift and new material gets sucked up from the mantle.

No, if another species on earth ever managed to learn to make tools and use them to fight each other over other resources -- they couldn't, because there's nothing left on the surface, not enough to make a difference, anyway.

The Earth will recover, as it did from the mass extinction that ended the Cretaceous. Life will, uh, find a way, to paraphrase Jurassic Park. But that's it for what we like to call "civilization."

I can't say the planet will be worse off for it. Sucks for us, though.

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