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 5 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 7 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.

July 26, 2022 at 12:02am

#1035699

A Crystal in Time

Oh, those crazy mad scientists...

What the heck is a time crystal, and why are physicists obsessed with them?
Some of today’s quantum physicists are tinkering with an esoteric phase of matter that seems to disobey some of our laws of physics.

You'd think scientists would learn from science fiction: don't mess with time crystals.

(No, this isn't a spoiler for the latest Star Trek series.)

You’re probably quite familiar with the basic states of matter—solid, liquid, gas—that fill everyday life on Earth.

And hopefully you're not intimately familiar with a fourth, plasma (not to be confused with blood plasma, which it might have been named after), which is a gas in a high enough energy state to be stripped of its electrons. Such as lightning.

But those three different sorts of matter that each look and act differently aren’t the whole of the universe—far from it. Scientists have discovered (or created) dozens of more exotic states of matter, often bearing mystical and fanciful names: superfluids, Bose-Einstein condensates, and neutron-degenerate matter, to name a few.

I always loved the phrase "neutron-degenerate matter." I always picture a bunch of drunk neutrons sitting around smoking and watching porn. But you know they're not actually doing anything illegal—otherwise they'd be charged.

I'll be here all week. Be sure to tip your server.

In the last few years, physicists around the world have been constructing another state of matter: a “time crystal.” If that seems like B-movie technobabble, it’s technobabble no longer.

Yeah, except in the movies, it's usually something that lets you see or travel through time. Don't be fooled by the name; this is nothing like that. Still cool, but not that universe-breaking.

In practice, that works something like this. You create a crystal whose atoms start in one state. If you blast that crystal with a finely tuned laser, those atoms might flip into another state—and then flip back—and then flip again—and so forth, all without actually absorbing any energy from the laser.

If you step back, what you’ve just created is a state of matter that’s perpetually in motion, indefinitely, without taking in any energy.

Okay, but you're still introducing energy in the form of the laser. That's not perpetual motion which, yes, is still fantasy.

That’s no small feat. It beats against one of classical physics’ most sacred tenets: the second law of thermodynamics. That law states that the amount of entropy, or disorder, always tends to increase.

Errgh. Wrong. What the SLoT says is that entropy (which isn't actually disorder) in a closed system always tends to increase or remain constant. I don't pretend to understand all the physics involved, but no, it doesn't shatter the Second Law.

The closed system thing is important, too. Idiots have used the SLoT to "prove" that evolution couldn't be possible because it would violate that principle, forgetting or ignoring that Earth's biosphere isn't a closed system; we get copious energy from the sun. This article is light on details (light? because laser?) but the crystal itself isn't a closed system either.

But these latest time-crystal-tinkerers did something different. They turned to Google and used a quantum computer: a device that takes advantage of the quirks of quantum mechanics, the seemingly mystical sort of physics that guides the universe at the tiniest scales.

Oh, it's Google? Then they'll probably yank it out from under us like they did Reader, Finance, and a bunch of other cool things.

Also, stop with the "seemingly mystical" garbage. Everything seems mystical until it's understood—though to be fair, there may be three people in the world who actually understand quantum physics, and I'm certainly not one of them.

So, could these time crystals indeed lead to a new wave of nascent time machines?

No. Look, I'm as big a fan of science fiction as anyone, but come on, stop with the bait. There is no reason to try to inject fantasy into what's already pretty cool. This is known as the "truth is stranger than fiction" principle.

But they might help make quantum computers become more robust. Engineers have struggled for years to create something that could serve as memory in quantum computers; some equivalent to the silicon that underpins traditional computers. Time crystals, physicists think, could serve that purpose.

And finally, an actual (if currently hypothetical) application. Not that physics needs an application, but it's good to know the real reason for the hype, which is not so you can go back in time and kill your grandfather before your mother was born, thus creating a world-breaking paradox.

To sum up, science is cool. Science fiction is cool. One should always heed the warnings of science fiction when doing science. But when you're writing about science, don't hype what ain't there.

If your brain won't explode, you can read more detailed information about time crystals here.

As that is Wikipedia, don't, like, use it for a dissertation or anything, but it's probably a decent overview.

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