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

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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

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WakeUpAndLive️~Happiness

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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

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

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https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

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

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WakeUpAndLive️~Happiness

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

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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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August 30, 2023 at 9:20am

#1054851

The Problem of Stupid

I've done entries about this topic a few times, so I had to double-check to ensure I hadn't linked this particular article before, or that it's not some variation on a previous article. And also to try not to duplicate a title. It looks like something different, though I can always make stupid mistakes.

Bonhoeffer’s “theory of stupidity”: We have more to fear from stupid people than evil ones
Evil is easy to identify and fight against; not so with stupidity.

I'm not so sure evil is so easy to identify and fight against, but I guess that compared to stupidity, it's a cakewalk.

The most relevant previous blog entry is here, from two years ago, if you're interested in comparing views on the subject: "Fight The Power"

Today's article isn't all that long, so this entry won't be, either.

There’s an internet adage that goes, “Debating an idiot is like trying to play chess with a pigeon — it knocks the pieces over, craps on the board, and flies back to its flock to claim victory.” It’s funny and astute. It’s also deeply, depressingly worrying.

I would like to once again draw a distinction between stupidity and ignorance. Ignorance is our natural, base state; it's only through learning stuff that we become not-ignorant. The key there is to be open to learning stuff, even if it sometimes contradicts stuff we've already learned. Ignorance is, therefore, forgivable, and fixable... if it doesn't turn into willful ignorance.

For theologian and philosopher Dietrich Bonhoeffer, the stupid person is often more dangerous than the evil one.

The core of theology and philosophy is to build logical edifices. Unfortunately, often those edifices have no foundation. Someone can be smart and still wrong, due to false premises.

In comic books and action movies, we know who the villain is. They wear dark clothes, kill on a whim, and cackle madly at their diabolical scheme.

The whole point to comic books and action movies is to have a clear villain, someone that the very clear hero can successfully thwart. It simplifies, and is therefore satisfying. More recently, both sides have been written to be on shaky moral ground, which can be satisfying, too.

Once something is a known evil, the good of the world can rally to defend and fight against it.

Except that it's blindingly obvious that this doesn't happen on a regular basis. You know Russia's still trying to invade Ukraine? Sure, we're helping Ukraine (the clear Good Guys), but not enough to win a definitive victory against the Bad Guys.

And propaganda has convinced some people that the Good Guys are the Bad Guys, and vice-versa.

Stupidity, though, is a different problem altogether. We cannot so easily fight stupidity for two reasons. First, we are collectively much more tolerant of it. Unlike evil, stupidity is not a vice most of us take seriously. We do not lambast others for ignorance. We do not scream down people for not knowing things.

I don't really buy this, either, and once again, let's not conflate ignorance with stupidity. There are words I struggle to pronounce, because I've only ever seen them in writing. Other people struggle to spell words they've only ever heard. Which of us is more ignorant? Neither, really. But I tend to be more judgmental of people who can't spell than of people who can't pronounce, because I can relate better. I'm aware of this flaw and can therefore work to mitigate it; certainly, I have others I'm not aware of.

Bonhoeffer’s argument, then, is that stupidity should be viewed as worse than evil. Stupidity has far greater potential to damage our lives.

I mean, I kinda see the point, and agree, to some extent. But none of this addresses the definition problem, whereby, for example, the extreme right thinks that the extreme left is stupid, ignorant, and evil, while the extreme left believes these things of the extreme right.

But stupidity is much harder to weed out. That’s why it’s a dangerous weapon: Because evil people find it hard to take power, they need stupid people to do their work.

This is why the trope of the criminal mastermind with brainless minions is perpetual. Think Lex Luthor and Otis in the 1977 Superman movie.

So, in short, I'm not completely convinced. This may be source-bias on my part (theologians in particular are always suspect for me), but, in keeping with the theme, I try to keep an open mind. I think that earlier entry's article presented a more cogent argument.

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