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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 15 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 21 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

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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