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 wanderlust-words-in-motion

about 23 hours ago
In response to "Revisited: "Science Is Almost Always Wrong...""

My ninth grade chemistry teacher used to threaten a kick to our Bunsen burner.

2 Like

🌖 HuntersMoon

a day ago
In response to "Revisited: "Science Is Almost Always Wrong...""

I feel for you, Bob. "Kiss my Pythagoras" never really caught on either.

2 Like

tj wanderlust-words-in-motion

a day ago
In response to "Bon Voyage"

Who knows, maybe they'll like the record, swap it out with one of their own and ship it back in the original container(Voyager), starting an intergalactic record exchange.

2 Like

🌖 HuntersMoon

Jan 18, 2025 at 10:14am
In response to "Bon Voyage"

It's not exactly like having your credit cards hacked.

For those interested...

2 Like

Elisa: Snowman Stik

Jan 18, 2025 at 9:31am
In response to "Bon Voyage"

"People like a happy-ending success story, I hear."

Now to see if this one gets a good response here or if it needs to be further shared.

2 Like

🌖 HuntersMoon

Jan 17, 2025 at 2:51pm
In response to "Up In Re-Volt"

Interesting... but the puns

@ Jayne: One of my all-time favorite scenes.

1 Like

tj wanderlust-words-in-motion

Jan 17, 2025 at 12:21pm
In response to "Up In Re-Volt"

After reading this I realized that most of what I know about electricity is that if you get a shock, it Hertz!

4 Like

Jayne

Jan 17, 2025 at 11:18am
In response to "Up In Re-Volt"

If only there was some way to milk cows with without electricity...

I know what they're referring to, but I still think it sounds dumb lol

3 Like

🐕GeminiGem🎁

Jan 16, 2025 at 6:15pm
In response to "Meat Me In the Lab"

I am a vegetarian for animal-rights reasons. I eat plant based meat, some are pretty good. The rest are an acquired taste. They can call cultivated meat "slaughter free" if they call the real thing "slaughter house delights" or something ridiculous like that.

2 Like

tj wanderlust-words-in-motion

Jan 16, 2025 at 4:18pm
In response to "Meat Me In the Lab"

In time, and once the bugs (possibly literal) are worked out, I think it's a great idea. Of course, twenty-five to fifty years down the road it's likely they'll find some kind of adverse health problems related to eating bio-meats, but if it's not that it would be something else. Life is like that and the answer isn't to not try, it's to push forward and overcome.

Will we (you and I) ever see the day every home has a food replicator in it instead of a microwave? Probably not, but it would not only solve a lot of food related issues, it would also solve a lot of problems of what to do with our waste products...

2 Like

January 20, 2025 at 9:32am

#1082596

With a Purposeful Grimace and a Terrible Sound

This article from Discover isn't very old, and it's about one of the most important international cultural figures.

Godzilla at 70: The Monster’s Warning to Humanity Is Still Urgent
The 1954 film 'Godzilla' launched a franchise that has been warning viewers to take better care of the Earth for the past 70 years.

Eh, if it'd been that urgent, we would have taken better care of the Earth for the past 70 years.

The 2024 Nobel Peace Prize has been awarded to Nihon Hidankyo, the Japan Confederation of A- and H-bomb Sufferers Organizations.

The Peace Prize is a joke and has been since it got awarded to a terrorist back in the 1970s. While last year's doesn't stoop to that level, I don't know how many have been awarded to an organization instead of an individual.

Fortunately, this article isn't actually about spurious awards; that was just what passes for a human-interest lede.

Around the same time that Nihon Hidankyo was formed, Japan produced another warning: a towering monster who topples Tokyo with blasts of irradiated breath. The 1954 film “Godzilla” launched a franchise that has been warning viewers to take better care of the Earth for the past 70 years.

A couple years back, a remastered Godzilla (more properly Gojira, from what I understand) got a screening in my local theater, in all its 1950s black-and-white glory. I canceled actual social plans to go see it, because it's Godzilla. I don't think I'd ever seen the entire movie start-to-finish until that night, and, from what I've been told, the version released in the US was severely edited from the original.

Point is, even if I had seen the whole film, it might have been the idiot version; also, I would have been a kid, and the whole metaphor would have been lost on Kid Me.

In our view, these films convey a vital message about Earth’s creeping environmental catastrophe.

Godzilla is as sudden and relentless as a nuclear weapon. The thing about a creeping environmental catastrophe is that it's, well, creeping. There's a parable about a frog in a kettle as the water slowly comes to a boil; it's inaccurate as biology, but still a decent metaphor for living in a crumbling ecosphere.

Superman's famous origin story comes to mind. In brief, Jor-El figured out that the planet Krypton was in imminent danger (details vary among retellings), but no one listened to him, so he sent his kid to Earth to become a god. When I was a kid, I was like "But why didn't they listen to Jor-El?" Well, now I know.

“Godzilla” is full of deep social debates, complex characters, and cutting-edge special effects for its time. Much of the film involves characters discussing their responsibilities– to each other, to society, and to the environment.

I've said that while American movies are about bad guys getting the shit kicked out of them by the good guys, Japanese movies are about honor. This tracks here, as honor is tied to responsibility. Another way to look at it is that American movies are about rights, while Japanese ones are about responsibilities. It's obviously not that simplistic in reality, and you can find all kinds of exceptions. (Meanwhile, French movies are about managing to have sex.)

Anyway, the article goes into how Gojira evolved from villain (sort of) to hero (sort of), and some of the other highlights and lowlights of the movie series. It pays special attention to the metaphors, but, really, there's nothing wrong with enjoying a good kaiju fight movie.

With a purposeful grimace and a terrible sound
He pulls the spitting high tension wires down
Helpless people on a subway train
Scream bug-eyed as he looks in on them
He picks up a bus and he throws it back down
As he wades through the buildings toward the center of town

3 Like • Comment

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