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.

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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.

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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.

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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.

May 19, 2023 at 10:37am

#1049836

Every Thing

You know why you should never trust atoms? Because they make up everything.

Why atoms are the Universe’s greatest miracle
With a massive, charged nucleus orbited by tiny electrons, atoms are such simple objects. Miraculously, they make up everything we know.

I could quibble about some of the words here, like "miracle," "simple," and "everything," but fine, they're not writing for scientists but to get ordinary people to know something about science. I can't be mad about that.

Similarly, I don't have much to say about the article itself. It's not technical, and it's got lots of cool illustrations, some of them animated. Highly recommended for anyone with curiosity.

One of the most remarkable facts about our existence was first postulated over 2000 years ago: that at some level, every part of our material reality could be reduced to a series of tiny components that still retained their important, individual characteristics that allowed them to assemble to make up all we see, know, encounter, and experience.

I mean, technically, light (which obviously enables us to see) isn't made of atoms. But it's generated from them, so okay.

Incidentally, I'm pretty sure that Democritus (the Greek who came up with the above idea) would be almost entirely forgotten had he not turned out to have been onto something. Lots of stuff the Greeks came up with didn't pan out (pun intended). And the Greek atomist theory wasn't exactly correct, either. It was more philosophy than science.

Everything that’s made up of normal matter within our Universe — whether solid, liquid, or gas — is made of atoms.

I could also quibble that this statement is a tautology, but why bother?

If all of human knowledge were someday wiped out in some grand apocalypse, but there were still intelligent survivors who remained, simply passing on the knowledge of atoms to them would go an incredibly long way toward helping them not only make sense the world around them, but to begin down the path of reconstructing the laws of physics and the full suite of the behavior of matter.

That's assuming they'd have the time to do so, between running from predators, finding prey, and hiding from aliens and/or zombies.

This is, however, just their framing device to communicate the idea of building atomic theory from the ground up. Best to not take these things too literally.

Like I said, I don't have much else to add. Mostly I just wanted to hold it up as an example of how one might communicate complex ideas to a wider audience.

2 Like • Comment • View 2 comments

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