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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10 Most Recent Blog Comments:

Wordsmitty ✍️

about 2 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?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

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

1 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

a day ago
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

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

June 6, 2024 at 9:51am

#1072234

Elevator to Heaven

From Wired, some panic about a thing that might not even happen.

What Happens If a Space Elevator Breaks
These structures are a sci-fi solution to the problem of getting objects into orbit without a rocket—but you don’t want to be under one if the cable snaps.

Yes, they're a staple of science fiction, but early concepts for them were thought up way back in the 19th century, before even controlled powered atmospheric flight and way before achievement of space travel.

They're also called Tsiolkovsky towers after the Russian who apparently first conceptualized them. According to Wiki,

other names include "space bridge, star ladder, and orbital lift." Personally, I find all those names descriptive but boring. No, I think it should be called a space lift. Because it rhymes with "face lift," and I find that amusing. Everything should be named in accordance with what I find amusing.

In the first episode of the Foundation series on Apple TV, we see a terrorist try to destroy the space elevator used by the Galactic Empire. This seems like a great chance to talk about the physics of space elevators and to consider what would happen if one exploded.

Not even the prospect of a Foundation series is enough to get me to pay any attention to Apple TV. But the idea, as I said, isn't new, nor is thinking about the security issues surrounding them. If you think about it, the idea of a tower connecting Earth to space will inherently offend a large portion of the population, who will inevitably call it the Tower of Babel, decide that it's against God's will, and actively try to blow it up.

This is why we can't have nice things.

Well, actually, we can't have a space lift because, so far, no one's been able to come up with a material that with the appropriate requirements, or address numerous other issues including powering the thing or shielding any passengers from space radiation. As far as I know.

Anyway, the article mentions that, sort of, but starts out by explaining, in great detail, how a space lift would be an improvement over rockets. This takes up most of the article, and I'm not quoting that part. Only after that does it get into what the headline promised.

In the first episode of Foundation, some people decide to set off explosives that separate the space elevator’s top station from the rest of the cable. The cable falls to the surface of the planet and does some real damage down there.

Again, not the first time this has been considered. I don't remember a scene like that from the Foundation books (by Asimov, whose entire output I devoured as a teen), but it's been a very long time. I do remember Kim Stanley Robinson addressing the issue in his Mars series.

The (over-simplistic, which the author acknowledges) video model in the article aligns with what I recall of Robinson's description.

So not only is building a space elevator very difficult, but you really don’t want the cable to snap and fall.

At this point, though, it's firmly in the realm of speculation, both in science fiction and actual science. The problem may be so tough that we'll actually develop antigravity first, which would negate the need for a space lift. Which is why you don't see many space lifts in Star Trek (budget considerations notwithstanding).

Of course, all of that assumes that a small remnant of humanity doesn't get plunged back into subsistence living before either happens, which I'm not prepared to rule out.

5 Like • Comment

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