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Monday, Nov. 18, 2024

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

tj-turkey-jobble-jobble-hard-J

about 13 hours ago
In response to "Revisited: "A Frank Discussion""

I became disgruntled with meat sold in local grocery stores; half the weight is water, the taste is bland, and it's "use by" date is usually one day away. I found a small town butcher/locker plant that sells locally raised meats and buy in bulk so it's actually fresher, cheaper, and better tasting. Recently, I decided to try my hand at making our own sausage as well. It's a bit of work, but I think it's worth the trouble.

1 Like

Turkey DrumStik

a day ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

Nov 14, 2024 at 12:10pm
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

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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