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 10 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

Sep 21, 2024 at 7:34am
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.

February 7, 2023 at 12:01am

#1044382

Pareidolia

Another entry for "Journalistic Intentions"

[18+]. This one can be found here: Faces Everywhere

Pareidolia

is popularly known as the tendency to see a face or figure in accidental places. Examples include the Man in the Moon, the Face on Mars (it's not a face), and the ever-popular Jesus toast.

Or, speaking of Jesus, some people find him in a dog butt.

I'm not judging. But it does confound the proverbial dyslexic agnostic insomniac, who stays up all night wondering if there's a dog.

In the case of today's prompt image, though... I'm not seeing it. Looks more like a grasping hand to me. Which is a form of pareidolia, but the title of the image led me to expect to see faces. Generally if I go into something with that expectation, I'll see it. Not this time.

This could be related to my partial face-blindness. A while back, I tried to get back into my bookface account so that I could delete it. I'd forgotten the password, so they issued me a challenge: identify the people in these photographs, ripped from my friends list.

Leaving aside the ones who I only knew from online, whose faces I wouldn't recognize anyway, I haven't seen some of my failbook friends in decades. And even absent those, I'm just not that great at identifying faces. Hell, one time I was at a bar doing a trivia contest. I had barely even started drinking yet, and the goal was to identify the famous people in a set of photos. I recognized a couple of them, but one that I didn't was... Bruce Springsteen, circa 1985. You'd think if I could recognize anyone, it's him. But no.

Long story short, this is why I still have a facebuck account: I can't get back in to delete it. And it's also why I don't always see pareidolia.

One thing that occurred to me a while back, though, is that artists rely on pareidolia. That is, I don't mean the examples in the Wiki link above, but, like, cartoon artists. They can throw together a few lines, and it looks like a general face (even to me). That's deliberate, though, and doesn't involve chance formations in nature.

Now, I could go into speculation about why we see faces everywhere, like the late lamented Old Man of the Mountain in New Hampshire, and make some guesses about how it was advantageous for our ancestors to think they saw faces. But I won't, because that's all it would be: speculation. The facts we know include that we do see faces or figures in natural objects. And sometimes that's fun.

But sometimes, it leads to wild conspiracy "theories," like with the Mars thing. Some alien must have carved it! Well, no, it's just pareidolia.

Which doesn't explain why there's a smiley face on Mars

. Do you see it there? That's the crater Galle in the Argyre Planitia. Might be easier to parse in this photo.

Okay, yes, it totally does explain it. Completely coincidental. Just because Mars is the only known planet to be inhabited solely by robots doesn't mean that said robots did that for our benefit.

Have a nice day!

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