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 7 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 10 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

about 21 hours ago
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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
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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"

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/

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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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December 18, 2023 at 10:24am

#1061132

Still Not Kidding

Getting back to article-sharing (I've had a few pile up on me whilst I was away), here's a complex issue that I'm sure someone out there has a simple solution for. It's wrong... but it's simple.

How millennials learned to dread motherhood
To our generation, being a mom looks thankless, exhausting, and lonely. Can we change the story?

I'll leave off with complaining about the "generations" nonsense for once, because that age range, as of right now, is 27-42; hence, discussions about having kids are mostly timely.

One more disclaimer: obviously, I'm a man, and I'm not presuming to tell women what to do or fully understand their experience. This article, however, was written by a woman. I'm not just assuming her gender, either; her bio is right there at the link.

I had been seeing my boyfriend for about a year, and though things were going well, we never talked about our feelings on having children.

I have no idea what the "rules" are now. I feel like "first date" is too early, but waiting a year seems... hazardous.

So one night in the summer of 2022, I finally asked him where his head was at.

He looked surprised, considering the question. “I think I’ve always wanted to be a father,” he said slowly, adding, “That doesn’t mean it’s a deal breaker, though.”

I can't really understand men most of the time, either. This could be a matter of "I really like the sex and I'm not ready to break up over this until you push the issue." Or it could be, benefit of the doubt here, that he was subject to society's expectations, which are still that "get married, have kids" is the default.

Still, I felt nervous and even a bit lonely, because I am not someone who has dreamed of being a mother; I’ve never particularly liked babysitting or even being around little kids.

One wonders if she expressed that to him. Not liking being around little kids is a damn good reason not to have any. Oh, sure, the aunties will be all like "it's different when it's yours," but that's not always the case.

One of the most viral TikTok videos last year, with millions of views and some 800,000 likes, is known simply as “The List,” featuring hundreds of reasons to not have children. (Reasons included: urinary tract infections during and after pregnancy, back pain, nosebleeds, and #89, “could be the most miserable experience of your life.”)

Gotta admit, I'm torn about this. On one hand, I despise DikDok with a burning passion. On the other hand, I agree with the sentiment; the downsides should be more clearly spelled out. On the gripping hand, I'm just suspicious enough to believe that someone behind TokTik might have a vested interest in reducing, or at least reducing the growth rate of, the US population.

I know some women who have decided to forgo motherhood altogether — not out of an empowered certainty that they want to remain child-free, but because the alternative seems impossibly daunting.

When you consider everything it takes to raise a child, compared to 100 or even 50 years ago, this shouldn't surprise anyone. For starters, you don't have help—in most cases, you don't get extended family or other close-by relatives to assist. Assuming a partnership or marriage situation, both partners are working outside the home, usually; this means paying for child care. And partnership is remarkably fragile, these days. When you factor in all the stuff you have to buy, feeding the baby-industrial complex, frankly, I don't understand how most people can even afford it.

Children used to be an economic boon; now, they're a liability.

On top of this, there is the well-documented aversion many millennials feel about making any sort of commitment, so conditioned are we to leave our personal and professional options open.

Job security, too, is a thing of the past. Most people job-hop; it's the norm, today. Without some confidence that your job is going to be, if not with the same company, at least in the same geographical area, life is uncertain. So, no, I don't blame people for not making commitments; it's hard to do that when no one will commit to you.

There's a lot more that I'm skipping, but let's pick back up at:

Previous generations “did not experience the same vocal outward world that we’re living in today where everybody is telling you it’s almost crazy for you to have children,” said Sherisa de Groot, founder of Raising Mothers, a literary group focused on parents of color. “That it’s selfish for you to have children. That it’s almost, like, a morally wrong thing to do at this point, because look at the hell basket we’re living in.”

I'm aware that a) we're not supposed to make fun of names and b) de Groot is a perfectly normal Dutch surname. But I'm thinking it. You know what I'm thinking, too.

Anyway.

I'm not saying it's morally wrong to have children. But one does need to consider that aspect, and decide for oneself.

One final point from me: The article quotes a study...

Eighty percent of respondents actually described parenting as enjoyable all or most of the time, while 82 percent said it was rewarding all or most of the time.

I won't go so far as to call that study bullshit, but consider all of the implicit bias involved... and, for someone still on the fence, it would be easy to draw the mistaken conclusion that this means you'd have a 4 out of 5 chance of enjoying the experience.

So, the article is fairly long, and raises valid points on all sides. I won't waste your time, or mine, further. Mostly, I just found it interesting that the discussion is still going on, and what it says about the state we're in.

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