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 20 hours ago
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🎁

about 24 hours ago
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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day ago
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

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

October 21, 2024 at 10:16am

#1078678

Mything the Point

From the BBC, an article at the intersection of science and mythology.

The myths that hint at past disasters
Myths and fables passed down over thousands of years are full of fantastic creatures and warring gods. But they also might contain evidence of environmental disasters of the past.

It's long been assumed that many, if not all, myths contain some elements of fact. (Like the article, I'm using "myth" in the sense of a passed-down story, not the more modern meaning of falsehood.) The tough part is teasing out the fact from the fiction.

The article opens with a bit about expected sea level rise, comparing it to past sea level rise. This might add fuel to the "climate's changed before and we're still here" campfire of climate change deniers, but so be it.

With the possibility of a catastrophic global sea level rise of 3ft (1m) by 2050 which could force millions of people to leave their homes, researchers have now started to look at ancient stories about land lost to the sea and downed cities in a new way.

Hell, it might have looked like the whole world was flooding, leading to one extraordinarily popular myth (though that one might have been due to its Mesopotamian origins).

Some researchers argue that tales of hot boulders thrown into the sea or the building of sea walls comprise factual information, albeit exaggerated and distorted to some extent.

Well, okay, argue all you want, but how about some science to back it up? I have this hypothesis that the "nephilim" mentioned exactly once in the Bible, in Genesis, are actually a passed-down memory of Neanderthals. There's no definitive translation to it, and the KJV muddled the waters by rendering the word as "giants," with even less evidence (we didn't know about Neanderthals at the time). Like I said, it's my hypothesis. I have literally no way of supporting or disproving it.

These researchers are geomythologists.

As good a name as any, and better than most.

“Geomyths represent the earliest inklings of the scientific impulse,” says Adrienne Mayor, folklorist, historian of ancient science and research scholar at Stanford University, California, and author of the important The First Fossil Hunters, “showing that people of antiquity were keen observers and applied the best rational, cohesive thinking of their place and time to explain remarkable natural forces they experienced.”

I suppose, for various definitions of "rational" and "cohesive," if such words can be applied to deciding that floods are due to gods yeeting boulders and whatnot.

There's a lot more at the article, including specifics; I won't quote much more.

Back when the glaciers retreated, humanity was already infesting the European lands (I'm including Neanderthals in that group). In that era, Britain wasn't an island; it was connected to the mainland by a low-lying chunk of land now called Doggerland. This wasn't particularly ancient, in geological terms; apparently, it flooded out about 6,000 years ago, around the time civilization was flourishing in Mesopotamia, thanks to beer.

I'd often wondered about undersea archaeology in that region, and what it might find in terms of artifacts. It's often called a land bridge, but that implies that humans mostly lived in Britain or Nederland, only crossing the "bridge," but there's no reason why people wouldn't have lived there.

Turns out, of course, that actual scientists were way ahead of me.

The whole thing's fascinating to me, though I doubt the article's thesis that there's much practical use in terms of dealing with catastrophic modern climate change in an era when there are billions of people, not hundreds or thousands, living near a coast. But finding more truth about ancient humans is practical enough for me.

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