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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Annette in France

about 9 hours ago
In response to "It's Hot in The South, Once Again!"

The quote is too small and low contrast to be read.

Elycia Lee ☮

about 19 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

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Wordsmitty ✍️

a day ago
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That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

a day ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

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Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

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Kåre Enga in Montana

Sep 25, 2024 at 7:53pm
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

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Annette in France

Sep 25, 2024 at 3:20pm
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

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Brandiwyn🎶

Sep 25, 2024 at 2:49pm
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

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Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

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August 6, 2019 at 12:02am

#963785

Prism Sentence

Today, we're going to discuss something that isn't.

https://www.theatlantic.com/science/archive/2018/02/what-color-tennis-ball-green...

What Color Is a Tennis Ball?
An investigation into a surprisingly divisive question

We think of color as an intrinsic quality of a thing. It is not.

I was stunned. I’d gone from being so sure of myself to second-guessing my sanity in a matter of seconds. More than that, I could never have imagined the question of the color of a tennis ball—surely something we could all agree on, even in these times—would be so divisive.

Well, someone hasn't been paying attention to politics. Probably best for them.

An investigation was in order. It began with the most obvious first stop, Wikipedia. “Tennis balls are fluorescent yellow at major sporting events, but in recreational play can be virtually any color,” the page for tennis balls said. And it’s true. A quick spin through Amazon shows that tennis balls, at least according to the labels, come in yellow and green and purple and even pink with Hello Kitty’s face on them.

TOO FAR.

“I make this decision as much on the basis of what I think I know about tennis balls—that they are yellow—as I do on what color I recall that they looked when I last saw one,” he said. “In other words, like the color of a lot of objects, how we label [a tennis ball] is determined both by perceptual and cognitive factors: the actual physical light entering your eye and ... knowledge about what people have typically labeled the objects.”

When I was a kid, I remember having a deep, philosophical discussion with a friend of mine (we did that sort of thing), and I pointed out that I had no way of knowing whether the color I saw and labeled as "red" (or any other color) was the same as what he saw and labeled as "red." Like, if I could see through his eyes with his brain, maybe I'd see blue instead, or chartreuse, or black. This blew his mind. Much later, this argument was presented on the internet as something earth-shattering. I really should patent these ideas.

Conway pointed to bananas as an example of this phenomenon. Bananas can be a number of colors—green when they’re not yet ripe, brown when we’ve let them sit out for too long. But we label bananas yellow, and we do it because “this is their state when we care about them,” Conway said. “It is canonical among most people to call bananas yellow.”

Yeah, okay, but when you look at an individual banana, you can assign it a color. Green, yellow, or (in the case of the ones attracting fruit flies on my counter right now), brown.

In other words, humans are good at pointing at a yellow paint chip in a line of colorful chips and saying, that’s yellow. But if we’re shown a yellow paint chip alone and asked what color it is, we become less certain about calling it yellow.

Like many cities, we have a "Yellow Cab" taxi service here. Some of the cabs are yellow. Most are orange. At least one of them has a silver paint job. This is annoying, so I use Uber.

Conway took it a step further, suggesting that the way people see tennis balls could reveal something about their lifestyles. Night owls, for example, spend most of their time under artificial, warm light, which means they’d discount warm colors and see a tennis ball as green.

Well, that explains why my initial reaction to the headline was, "Green. Duh."

So now I'll go back to my original assertion: Color is not an intrinsic quality of a thing.

No, this doesn't mean it's an "illusion." I'm sick of people labeling things that are perceived differently by different people as an "illusion." Like, "Time is an illusion." No it fucking isn't. Yes, we each perceive it differently - mostly this is psychological and/or age-related; relativistic effects are really minor for us earth-bound clowns. But they're real. So is time. So, no, color isn't an illusion, but it does depend on certain quantum effects as well as one's perception.

I mean, we can talk about "the green grass," and I think most people associate green with grass. The only exceptions I know of are RG colorblind (duochromatic, technically), but even they have been inundated with social expectations like "the grass is green."

So imagine my horror, as a kid, when I heard about Kentucky bluegrass. No, not the music; the actual grass you can put on your lawn. I finally tracked some down at one point and scratched my head. Looked green to me.

I can't be arsed to get into the technicalities of it, but grass is green because that's the only color reflected from it when enough light of the right spectrum (e.g. the harsh glare of the accursed daystar) shines upon it. This has to do with electron orbits and quantized energy; like I said, not getting into it.

Point is, shine light of a different spectrum on grass, and it could appear to be another color. Or even if it's moonlight, which is highly attenuated sunlight reflecting off dark gray rock, and grass becomes gray or something darker.

And yes, the moon is mostly colored dark gray. That brings me to the other point about color, which is that it looks different depending on what other colors are around it. This is even less an intrinsic property of the thing, and more a matter of human nerve cells. Take a look at these, for example:

http://mentalfloss.com/article/54448/5-color-illusions-and-why-they-work

There are many other examples; a quick Google search will root them out.

So, yes, color is somewhat subjective. And arguments about it are idiotic. Unless something is colored Hello Kitty pink, in which case, AAAAAAH MY EYESSSSSS.

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