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 11 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 14 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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October 15, 2023 at 9:53am

#1057429

Revisited: "Headline Questions Are Usually Answered 'No'"

Reaching back to August of 2019, I landed on this entry: "Headline Questions Are Usually Answered "No.""

It contains a raw link (not an xlink) to a Nautilus article. The link no longer works, but I guess they simply switched around the way they handled URLs in the intervening years, because a quick search found the original article. Here it is in my more current format:

Does Depression Have an Evolutionary Purpose?
Some psychologists believe suicide and depression can be strategic.

Now, the article itself is from even further back in the past: February of 2017. So the field might have experienced some changes since then, especially what with a few years where a lot of people, if they could get psych help at all, did it remotely. Which seems to me like a recipe for disaster, especially for extroverts, but what do I know?

So, I'm still not going to cherry-pick quotes from the article; as I said last time:

Now, usually, I mine quotes from the articles I link, but few of them in this one are really worth isolating; I think one needs to read the article to get the idea.

This, then, will focus on some of the things I wrote back then in the Before Time.

I have problems with how evolutionary "explanations" for this and that and the other thing are generally portrayed.

Speculation from evolutionary psychology has become so pervasive in human biology reporting that, unless the article in question is especially compelling, I just quit reading when I get to lines similar to: "This makes sense from an evolutionary perspective. In the distant past, our ancestors would have..."

Because it's almost always speculation, and it almost never takes into account that our evolved traits, whatever they may be, include holdovers from even more distant ancestors than our lonely great^great-grandparents on the African savannah.

They're origin myths, "Just So" stories, only written to appeal to a slightly more scientifically literate audience. The only difference between them and "God made us that way" is a lack of supernatural references, an acknowledgement that we are, in fact, products of evolution. Which is better, but still not great.

With evolution, not every trait is a survival trait. Some are vestigial or effectively so. Others are incidental. One way evolution works is that incidental traits sometimes end up aiding survival and/or reproduction, so those traits can get passed on. Vestigial traits like - I want to say the appendix, but I've heard that might actually be part of the immune system - whether or not you can wiggle your ears are generally neutral to survival, but might have had some benefit in a distant ancestor.

In that case, by "distant ancestor," I meant pre-hominid.

As for the appendix thing, since then, I've heard that the appendix does indeed have a function: as a store of beneficial gut microbes. When the intestines are subject to disease or poison, disrupting the flora there, it's supposedly meant to pump the good bugs back into the tube.

Now, it's not like whoever told me that was a doctor or even a biologist, so I wouldn't take that as Absolute Truth or anything. From a purely anecdotal perspective, I never had big problems with weight control before the appendectomy I endured back in the early noughties, but that could easily be correlation without causation.

I mention this because there also seems to be a correlation between gut microbe health (or, more properly, lack thereof) and mental health, so it might actually be relevant to the article. Or it might not. I don't know.

Also I should add that while I've been depressed, I've never been suicidal; I don't know what the stats on that are, but I think the article conflates "depression" and "suicidal ideation," and that pisses me right off.

Still does. I'm still subject to the occasional round of depression, and the closest I've ever come to contemplating suicide has been from a place of writer's curiosity: if I were to write a character deliberately ending their life, how could I do it in a way that seems realistic to readers?

From that perspective, and no others, I've also contemplated such things as rape, theft, murder, war, and what it would be like to be Southern Baptist. Stuff I'd also never attempt. So I don't think it counts

Still, the article makes some interesting points and I'm curious to see if this line of speculation goes anywhere.

Despite my gut feeling (pun intended, of course), there could be something to the "evolutionary psychology" explanation. But there needs to be more than just guesses, though I have no idea how one could go about doing the requisite science to support or falsify such a hypothesis.

One final thing: the end of the original article provides a link and a phone number for suicide prevention. I'm not 100% sure, but I think they've changed the phone number so you can get there by calling 988.

Which makes me glad we don't use rotary phones anymore. I can just imagine someone getting to the second digit, waiting for that goddamned dial to spin back around, knowing there's still another one to endure, then going "fuck it" and blowing their brains out.

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