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 13 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 16 hours ago
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With all that walking, I hope you stayed hydrated.

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Annette in France

a day ago
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Bienvenue en France.

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WakeUpAndLive️~Happiness

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

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WakeUpAndLive️~Happiness

a day ago
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Elisa: Middle Aged Stik

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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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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September 1, 2023 at 11:11am

#1054947

Let's Settle This Like Men

Whenever two people need to decide who's doing what undesirable task, they can always figure it out with fists.

The Psychological Depths of Rock-Paper-Scissors
An excerpt from veteran game designer Greg Costikyan's book "Uncertainty in Games."

As the subhead suggests, this is a book promotion. Whatever; I still enjoyed the article.

Unless you have lived in a Skinner box from an early age, you know that the outcome of tic-tac-toe is utterly certain. At first glance, rock-paper-scissors appears almost as bad. A four-year-old might think there’s some strategy to it, but isn’t it basically random?

I'll note here that the website this comes from is The MIT Press Reader, and the book it promotes is in the field of psychology, so they can be excused for expecting their audience to know what a Skinner box

is.

Players quickly learn that this is a guessing game and that your goal is to build a mental model of your opponent, to try to predict his actions.

So, it's good practice for poker.

Note the use of the masculine pronoun. While it's a game anyone with at least one mostly-intact hand can play, I've never heard of women settling their differences in this manner. I'm sure it happens; it just seems to be primarily a dude thing.

The reason rock-paper-scissors is not a purely arbitrary game, and the reason that an excellent player will win more often than chance would predict, is that human psychology is not random, and some behaviors are — not necessarily predictable, but likely to occur more often than chance would dictate.

This is also what some people miss about classic cheating methods such as loaded dice or marked cards: they give the cheater an edge; they're not going to work 100% of the time. That would be too obvious, anyway. But even if you can shift your odds from 17% to, say, 24% (not unreasonable for loaded dice), you still do better in the long run.

The article, obviously, goes into some of the psychology involved. It seems to be related to the Gambler's Fallacy, only kind of in reverse? Even if you have no intention of ever making decisions via RPS, the analysis has other implications.

In other words, at first glance rock-paper-scissors appears to be a guessing game, with victory going to the player who can outguess his opponent; at second glance, it appears to be purely arbitrary; and at third glance, the original supposition is justified. It is, in fact, a guessing game with victory going to the player who can outguess his opponent, but there are strategies to “outguessing.”

If that makes your head hurt, consider the RPS extensions that have been developed by dedicated RPS researchers all over the world. Perhaps the most famous one is Rock-Paper-Scissors-Lizard-Spock,

popularized by the show The Big Bang Theory.

I've only seen a few clips from that show, enough to know that it's the nerd equivalent of blackface. RPSLS is still not complex enough for true nerds. For that, we need other extensions.

Like this one.

Or this one, with a brain-melting 101 possible throws.

You're welcome.

3 Like • Comment • View 3 comments

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