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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QPdoll

about 13 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 16 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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September 5, 2023 at 10:21am

#1055201

The Man, The Myth, The Reality

Some things we think are real turn out to be myths. But sometimes it's the other way around. Cracked provides a few examples.

5 Apocryphal People That Were Actually Real as Hell

Hell, of course, is not real (except for the town in Michigan), but I suspect they mean that as an intensifer.

Of course, embellishments are natural, but the basis for what you might think are tall tales are often a flesh-and-blood fellow.

Those "embellishments" might be how myths and legends get started in the first place. Kind of like how that fish you caught that one time keeps getting larger and heavier every time you tell the tale, until everyone's convinced that you hooked Leviathan. Or, well, you're convinced they're convinced, anyway.

5. Johnny Appleseed

I knew this was a real guy; I only ever objected to his glorification. Though I suppose his legacy of spreading non-native species around is better than that of Charlie Kudzuseed or Frankie Freshwatercarpegg.

Johnny Appleseed is a tale that sounds patently insane. A man wandering the Wild West, tossing out apple seeds and leaving a trail of trees in his wake, sounds like a folk tale at best and a severely deranged man at worst.

Well, he was at base a missionary, so "deranged" isn't too far out there as a description.

4. John Henry

As far as American myths-based-in-fact go, this one's my all-time favorite.

Heck, John Henry versus the machine is practically writers’ struggle against advancing A.I. Well, if John Henry was a feeble, depressed man, and the steam machine made weird-looking, factually inaccurate railroads.

The thing people get most wrong about this one, though, isn't the human vs. machine thing, but the nature of his work: he didn't hammer spikes into ties (well, he may have done that, too, but that's irrelevant to the main story), but, as I understand it, he drove metal drill thingies into raw rock for placement of demolition charges. You know, to make cuts or tunnels in the Appalachians.

3. Paul Bunyan

Rarely has there been a better modern example of reality getting exaggerated into mythology.

Bunyan’s physical legend, it’s believed, comes from a French-Canadian man named Fabian Fournier. Remember, first, that in 1875, when Fournier lived, the average male height was right around five and a half feet tall. Given that knowledge, the six-foot-tall Fournier was practically Shaq with an axe. It’s thought that Fournier’s famous bulk combined with the name of another famous lumberjack, Bon Jean, which over time and error, morphed into Bunyan.

So, roughly contemporaneous with John Henry. I'm seeing a lot of Johns and (French) Jeans on this list. Where "Paul" came from is unclear, but I guess it played better with test audiences than "Fabian."

2. Davy Crockett

I didn't think there was doubt that Crockett was an actual person. Still, sometimes it's hard to separate the truth from the exaggeration.

1. Dracula

Hey, someone not from North America.

He is, however, based on a real, notably horrible man. Vlad the Impaler, born Vlad III Tepes, was a ruler of Wallachia, which is now Romania.

Pretty sure the the Dracula thing is widely known to be based on a real, if morally repugnant, individual. But how did he get all the attention, while Elizabeth Bathory,

arguably a far scarier individual, languishes in relative obscurity?

Probably a sexist thing. After all, you don't see any chicks on this list, do you? If I had time, I'd come up with some.

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