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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Adherennium Plotting something

about 2 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.

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').

1 Like

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

1 Like

May 30, 2024 at 8:09am

#1071890

Job Security

From Cracked, an example of how we can understand and not understand science at the same time. Kind of a superposition of states, like an unobserved particle. See? I can science metaphor. It's wrong, but I can do it.

4 Scientists Who Only Added More Mystery to the World
Their problem-solving skills are only rivaled by their problem-creation skills

The misunderstanding is that scientists solve problems. I mean, sure, you get some answers, but those answers always lead to more questions. This is good, though; it's job security.

You want actual problems solved? That's what engineers are for.

There’s a bit of an erroneous belief that in order to become a famous scientist, you have to solve problems.

At least the author admits that it's a misunderstanding.

However, there’s another, arguably vastly more annoying way to get your name on an enduring thought. That’s to come up with a brand new intellectual mess for all the other scientists to have to try to figure out.

Like I said. Job security.

4. Thomas Young

I will admit that I've either never heard of this individual, or forgot that I did.

In 1801, Thomas Young disagreed with the popular belief that light was made up of particles. He believed that light was, in fact, a wave, and so he cooked up an experiment to prove it. He cut two slits in a sheet of metal and shone light through them.

Oh, yeah, the double-slit experiment. I've certainly heard of that. Hell, in physics lab in college, we performed it, only we used lasers (which hadn't been invented yet in 1801). It's clearly more famous than its creator.

Young set out to perform a simple experiment with light, and ended up creating what would be called “the central mystery of quantum mechanics.”

Sometimes, complex questions have simple answers. This is a case of the opposite. The article goes on to explain exactly why, and, miraculously, it conforms with my prior knowledge (that is, I'm sure an actual physicist could pick it apart, but for a comedy site, it's remarkably accurate).

3 Fritz Zwicky

This one, I'd heard of.

You might not know the delightfully named Fritz Zwicky, but you have heard the two words he coined in combination: dark matter.

Thus also providing job security for science fiction writers who are free to give it all kinds of magical properties.

Dark matter, which is — keep in mind that as an art major, I am fighting for my life here — matter that contains mass but emits no light and therefore cannot be observed, was his best, confident attempt at making some very nonsensical measurements make sense.

I gotta say, I'm impressed that an art major got so many things right. No disrespect intended to art majors, but they're not known for understanding physics. On the flip side, I have some small understanding of and education in physics, but I can't do art to save my life, so it all balances out.

Anyway, in my own amateurish way, I tend to see the concept of "dark matter" as a placeholder, a concept expected to have certain properties. Kind of like the luminiferous ether proposed before we more fully understood the nature of light (as per #4 above). When we finally figure it out, I'd predict that "dark matter" will be an historical relic, like luminiferous ether or the humour theory of medicine.

2. Enrico Fermi

You know this guy developed other stuff, right? Like, his famous "paradox" (which I've insisted in the past is not an actual paradox) was kind of a footnote to an incredibly productive career? And that he has a whole class of subatomic particles named in his honor?

Honestly, I think if he hadn't been so prolific, no one would have paid attention to the "paradox."

Nobody was arguing that the question “does extraterrestrial life exist” was too easy to answer. Yet, a man named Enrico Fermi decided to add another layer of unsettling confusion to that little gray layer cake, just in case anyone was feeling they had a good handle on it. Even worse, he reportedly rattled off his new addition casually at lunch, and every scientist since has been dealing with his bullshit. Bullshit that’s most commonly referred to as the “Fermi Paradox.”

It's nice to see my opinion on the matter confirmed, even it is by an art major writing on a dick joke site.

The article, again, does a good job explaining the questionable conundrum, so I won't repeat it here.

1 The Guy Who Invented Mystery Flavor Airheads

That wasn't a scientist. That was a marketer who probably got ordered to find something to do with the excess Airhead slurry left over after batches had been produced.

And we don't even know if it was a guy.

4 Like • Comment

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