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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Writer_Mike

about 8 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

a day 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.

2 Like

Annette in France

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

Bienvenue en France.

1 Like

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

1 Like

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

1 Like

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

2 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:46am
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

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

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

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
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.

March 27, 2023 at 10:06am

#1047040

On the Bias

A prompt for "JAFBG"

[XGC]...

What's your opinion on bias in the media?

There's not enough of it.

Allow me to explain before jumping on me:

Every media outlet, indeed, every individual, has bias. Some of it is conscious; some of it isn't. With media outlets, some of them have an obvious slant in one direction or another. Others attempt to be neutral, though even choosing which stories to run shows some bias.

And it's in that attempt to be neutral that, sometimes, you run into problems.

Let's take a possibly extreme, fictional example. I'm totally cribbing this from Superman:

Say there's an independent scientist who discovers that the planet's about to blow up. He brings this to the planetary council. The planetary council, who all have a vested interest in the planet not blowing up, ignore his findings despite overwhelming evidence. So he takes it to a nice, objective, attempts-to-be-unbiased news distributor, presenting all of his scientific evidence and his argument for why the planet is going to explode.

The media outlet promptly checks his story with the council, which scoffs at the very idea. So the journal, in an attempt to cover "both sides of the debate," meticulously prints the opinions of the independent scientist, who had evidence, and the council, who basically just don't want the planet to blow up (not that I blame the council for that).

This gets out to the average Kryptonianmedia consumer, who probably doesn't understand all the science involved, and pretty soon the planet's divided in two parts (for a few days before being divided into a billion parts): one, we trust the scientist because he presented decent evidence, and two, we trust the council because, well, because they're the council, I guess.

There are the inevitable follow-up stories, interviews with the folks on the street. "What do you think of the claims that the planet's about to blow up?" "I think we should take this seriously." Or, "I think it's a load of hogwash" (Or whatever the inhabitants of that planet used to clean their equivalent of pigs.) (Yes, I know that's not where "hogwash" comes from.)

So, here's the problem with those media outlets: they're treating informed, scientific findings with the same weight as uninformed, wishful thinking, which in turn acquires the same weight as the blithering of anyone who has a vested interest in the matter.

Now that I think of it, maybe this was the real reason Clark Kent became a reporter.

But I digress. The point is that, in their attempt to be perfectly fair and unbiased, the news outlets did their planet a disservice, resulting in the planet blowing up and there being only one survivor (or two, or a whole bunch, depending on which timeline we're talking about... sorry, digressing again).

Now, sure, you get some old bearded guy with signs off the street who keeps proclaiming, "The world's about to end," and clearly, you either a) want to see his evidence or b) want to get as far from him as physically possible. But someone who researches this sort of thing? Maybe they have a point. Maybe ask other people who research these things, and not Bil-Bob in front of the weed dispensary. Sure, scientists can be wrong, but the ones to make that assessment are... other scientists, not politicians or social workers. Or, for that matter, journalists.

All of which is to say that if you have an article about how the Spring Solstice happens because of axial tilt, don't think you have to interview a flat-earther for an "alternative viewpoint."

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