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

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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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WakeUpAndLive️~Happiness

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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

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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

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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

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WakeUpAndLive️~Happiness

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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 21, 2023 at 9:29am

#1057789

The Optimism Trap

It's a five-year-old article, but what the hell; it's new to me. From GQ:

Why Your Brain is Wired for Pessimism—and What You Can Do to Fix It
We’ve evolved to expect the worst. Dr. Martin Seligman, a psychologist, explains why—and what you can do to get some optimism back.

Why in the ever-loving shit would I want to fix what ain't broken? How can I get back what I never had? And most of all, am I going to have to once again rage against the misuse of evolutionary psychology in articles?

And why do I bother with GQ if I know I'm going to rant about it? Okay, that one I can answer: because it's fun.

Ever had someone tell you to just cheer up? Did it drive you crazy? Well, turns out that someone telling you to “be happy” isn’t just annoying—it’s also wildly unhelpful.

It's especially unhelpful for people with clinical depression.

Seligman compares being happy to falling asleep: it’s not something you can actively do—in the way you can get stronger by lifting more weights. It just kind of has to happen.

Okay, sure, but unless you're tired past the point of exhaustion, the way to fall asleep is to get into a comfortable position and pretend to sleep until, at some point, you either fall asleep or say "screw this" and go play a video game.

My point is that in order to sleep, we usually first have to act like we're asleep. In that analogy, we'd have to pretend to be happy in order to be happy. So, do you still want to compare being happy to falling asleep?

Now, yes, I've said in here at least a dozen times that happiness isn't a goal, but a byproduct. So, sure, I don't completely disagree with that premise; I just had to nitpick the analogy.

And as the father of positive psychology—the study what makes a good or meaningful life—much of Seligman’s work has dealt with trying to help people figure how to make it happen.

Hurk.

“Half the world is on the low positive affective spectrum,” he says referring to positive affectivity, a trait that usually correlates with sunnier dispositions. “I'm part of it, and a lot of the justification for what I work on, and what I write, is to try to help half the world, who is not naturally positive affective, to be more positive and optimistic.”

WHY?! OH GODS WHY?!!!

What if... what if the only people who benefit from someone being mindlessly optimistic are 1) those around them who don't have to deal with someone's annoying pessimism and 2) our overlords, who will more easily control a populace who believes the best will happen?

What he has learned is that well-being can be broken into five elements: positive emotion, engagement, relationships, meaning, and accomplishment (PERMA).

Boy, shrinks love their mnemonic acronyms.

Why does it seem like we are wired for pessimism?
The species that [was] going through the Ice Ages had been bred, and selected, through pessimism.

Not only is that abysmally terrible evo-psych, but it's easily falsifiable terrible evo-psych. Not that I've disproven it, mind you; but consider: not all of humanity was directly affected by an ice age. Many populations were equatorial or near-equatorial. Are their descendants happier today, controlling for all other variables?

But the main reason why it's horrible evo-psych is that, as usual, it assumes that the only evolution that mattered to us started with humans, instead of us having ancestors dating back to the dawn of life, each of which contributed factors to evolution.

So is this at odds with something like mindfulness, which argues you should be present in the moment? If you're focusing on optimism, you're also sort of missing the present moment, right?
Well, I think, if you look at what people are doing, and what you're doing right now when we're talking, you're prospecting into the future.

But if positive psychology is at odds with mindfulness, and I also despise mindfulness, oh no! Cognitive dissonance! Wait, no, I've got it—I can hate both.

I've read a few people have said that you might be better off cultivating a sort of non-attachment to well-being: be mindful that a lot of life is going to be suffering, and if you can find contentment in that, you might be better off than seeking out happiness.
I think the good thing about meditation—mindfulness, concentrating on the present, detaching—is as good anti-anxiety, anti-anger tools.

"A few people have said," interviewer? A few people? You do know that that's Buddhism, one of the most widespread spiritual practices in the world, don't you?

Of all the things you've studied, or learned, is there one idea you constantly find yourself encountering most frequently?
I think it's hope.

Despite all my rage, I am still just a rat in a cageI do believe in hope. I'll leave off quoting the article and explain that now.

So, let me simplify things, but with a simplification that can easily be extended to our more complex lives:

You have two possible outcomes, A and B. Maybe they have predefined probabilities, like drawing a certain card from a shoe in blackjack, or maybe not, but it doesn't matter. Let's say you label A bad and B good, or at least less bad.

From what I understand of positive psychology, it tells us that you should believe that B will happen. You should manifest that B will happen. But then if A happens, which it still might because no amount of manifesting will change the fact that it can happen, you're crushed, devastated, forlorn, lost. Whereas if A happens, you might feel a fleeting jolt of accomplishment, serotonin momentarily coursing through your neural network, and then it's gone.

On the other hand... if you convince yourself that A will happen, if you predict A, if you act as if A were the only way that the universe could possibly work... when A happens, you're not nearly as devastated, because you expected it. Whereas if B happens, you're not just experiencing fleeting pleasure, but absolute joy.

In other words, being pessimistic, seemingly paradoxically, must lead to greater overall happiness. But that's probably only true if you still hold out some hope that B might occur; and that's what I mean by "I do believe in hope."

This philosophy is most usually expressed by "expect the worst, but hope for the best."

No, I'm not deliriously happy all the time. But I have something I consider far more valuable: contentment. I may not be where I envisioned myself when I was younger, but I'm doing okay.

And that's the real trap of optimism: you think you can always do better, so you strive, you make changes, you expect things to improve, and then you're discontented when they don't. You're disappointed that you don't have what you want, instead of being satisfied with wanting what you have.

Or hell, I don't know. Maybe optimism works for you. I'm not judging. I only take issue with the idea that everyone should strive for some nebulous, glorious state of "happiness" at all times.

Incidentally, this is not the last GQ article in my queue. And the other? Well, it's even worse...

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