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

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

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Wordsmitty ✍️

about 18 hours ago
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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

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WakeUpAndLive️~Happiness

a day ago
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WakeUpAndLive️~Happiness

a day ago
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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"

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WakeUpAndLive️~Happiness

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

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WakeUpAndLive️~Happiness

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

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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 22, 2023 at 10:23am

#1054434

Boregon Trail

After that bit of head-stretching yesterday, here's something a little closer to home:

Top 10 States Voted as the Most Boring to Visit in the U.S. by Americans

Admittedly, "boring" is subjective. By what criteria? Are your expectations too high? Are you a teenager and find everything boring? Do you live in a big city and wish someone would mug you more often? Well, I've been to all but three states in the US, so I obviously have my own thoughts on this list.

1. Nebraska

According to the next traveler Nebraska is the most boring state in America, and they stated that there is nothing that would motivate them to visit again.

And naturally, this is one of the three states I haven't been to. But assuming it's an average of Kansas and South Dakota, sure, traveling across it probably induces boredom..

They seemed to believe that the state lacked excitement and allure compared to other states. Many fellow Americans agreed, and another stated, “There is nothing but cornfields.”

What? You don't like to eat? You don't like to drink bourbon? You have some deep-seated hatred of farming culture, despite, you know, depending upon it?

I can believe Nebraska is boring to drive through. However, statements such as these say more about the urban ennui of the statement-makers than about the subject matter.

2. Delaware

Oh, come on. You just included this one so people couldn't go "that's why they call them flyover states."

The commenter wrote that all three counties of it are nothing but an elaborate tax dodge masquerading as a state.

That bit is legitimately funny. Like, three counties in a trench coat pretending to be a state.

3. Kansas

Yeah, well, see above.

4. North Dakota

I admit I had to play little games with myself while driving across North Dakota in order to alleviate the boredom, but the state has a unique beauty to it. That probably extends across the border into southern Manitoba and Saskatchewan, but I have no first-hand evidence of this.

The worst thing about ND, from my perspective, was trying to find a place to crash. There was an oil boom going on while I was there, and everything was full.

5. Iowa

I... kinda love Iowa. Not the politics or anything, but the scenery. It's not nearly as flat, or as uninteresting, as people make it out to be. Maybe this is because I don't use interstates on my road trips.

6. Ohio

Despite being the birthplace of many famous individuals, such as the Wright Brothers and Neil Armstrong, this traveler believes that Ohio is lackluster regarding things to do and see. The user argues that Ohio is simply a state to pass through rather than a destination worth stopping at. They acknowledge Ohio’s impressive record of producing more astronauts than any other state but ultimately deems it insufficient in excitement and interest.

Oh, fuck right off with that. You want excitement? Stay in NYC and L.A., and leave the rest of us alone. We don't want you, either.

On the other hand, it could be that so many astronauts come from Ohio because they felt a deep-seated desire to get as far away from the state as possible...

I won't quote the rest, other than to say that one of them, Michigan, is also one I haven't yet visited. (The third one I'm missing is Alaska.) But it's clear to me that some of these responders have no sense of adventure, or beauty. Despite my gripes above about driving across certain plains states, fact is, I'd do it again. Sure, I have a much higher tolerance for what others consider boring, but in practice, it's a beautiful country with lots of hidden gems scattered across it. I'd do it all again.

Hell, even getting stranded in South Dakota a couple of years back ended up being quite the adventure.

In short, if you find one of these states boring in general, that's on you, not the state.

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