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 15 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 18 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/

3 Like

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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August 10, 2023 at 9:53am

#1053910

In Retrospect, These Were Bad Ideas

Today's list, from Cracked, is way too short. It fails, for example, to tell the story of "Mad" Mike Hughes, the flat-earth rocket scientist who, in an effort to see for himself the flatness of the planet, built a steam-powered rocket in the wilderness of Southern California, and promptly died in a lithobraking maneuver. I wrote about him, last year, here: "Bad Advice"

So here's more of that sort of thing, albeit not nearly as steam-powered:

4 Stubborn Idiots Who Were So Wrong That They Died

Stubbornness isn’t a particularly delightful trait to interact with.

It is also not a virtue, no matter how stubbornly people insist it is.

Dealing with a bullheaded friend or acquaintance is likely to end with a prominent forehead vein and a conversation that concludes in “okay, whatever, I don’t even care anymore.”

When a Gen-X meets a Taurus...

Combine that same stubbornness with an incorrect belief that’s considerably higher stakes, and you’ve got a recipe for disaster.

Like crashing your homemade rocket in the California desert. With you inside. Or like these items:

4. Franz Reichelt

For whatever reason, I've seen this guy from multiple sources recently.

Franz Reichelt was highly interested in the human power of flight, a hobby that, throughout history, has a pretty abysmal survival rate. Reichelt, however, lived at a time where airplanes already did exist and functioned — mostly. His interest took a slightly different slant, which was the development of a parachute suit that would allow pilots, or thrill-seekers, to jump from a great height and float down safely to the ground. To give you some idea of how this went, I can assure you that Reichelt is absolutely not the inventor of the parachute.

Or, for that matter, the wingsuit.

After what had to be an incredibly irritating back and forth with French police, Reichelt got permission to toss one of his dummies off the top of the Eiffel Tower. When he arrived at the top, though, he committed one of history’s most ill-advised and disastrous switcheroos, deciding he would wear the suit and jump himself.

And that, mes amis, is why today we call it the I-Fell Tower.

3. Herman Cain

As far as jokes about stubbornness go, politics is rather low-hanging fruit. But this guy was an entire branch of low-hanging fruits.

No person became more of a grim figurehead of the health risks of being a huge asshole than ex-presidential candidate Herman Cain.

I guess he wasn't Abel to let go of his stubbornness.

2. William J. Bailey

Throughout history, humans have come up with no shortage of wackadoo medical treatments and various snake oils.

Snake oil, it turns out, has legitimate uses. Most of the things we label "snake oil" are just useless. Some might actually do something, but no one's done studies. They don't make up for the ones that, you know, kill people.

In the 1920s, for example, radium was all the rage. Yes, the radioactive kind. Looking back, you can find it casually included in all sorts of consumer products like makeup or the glowing numerals on a watch face. Maybe the most straightforward, and incredibly stupid application was invented by a man named William J. Bailey, who created and sold Radithor, an “energy tonic” that was created from the simple recipe of water plus radium.

I'm not an expert on radioactivity (there are different kinds of radioactivity, and some of them are relatively harmless if their sources stay outside your body), but my understanding is that ingesting radium is particularly harmful because the body's chemical processes think it's calcium. So you end up with radioactive bones.

Suddenly, those glow-in-the-dark skeletons you see around Halloween aren't so cute, are they?

1. The Titanic

This one's so famous that I don't know why they even bothered including it on this list.

It might not be a single person, but I don’t think we could really end this article without acknowledging one of the greatest acts of human overconfidence in history.

This is true, but few of the people involved were directly, corporeally affected.

The submersible that went to look at the wreckage earlier this year, though...

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