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Saturday, Dec. 21, 2024

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Complex Numbers

by Waltz Invictus

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-Merry Mischief Maker

about 17 hours ago
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

about 22 hours ago
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

about 22 hours ago
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

a day ago
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

🌖 HuntersMoon

Dec 19, 2024 at 8:53am
In response to "Five Seconds"

2 Like

Aiva Raine

Dec 17, 2024 at 7:39pm
In response to "Need for Speed"

Even with the actual explanation- it still left me going- sooooo what's the difference between the special expensive "speed tape" they use for "temporary fixes" and duct tape? LOL- sure sounds like duct tape to me and they are using it for exactly the same reasons you would use duct tape. I'm guessing that the only difference between it and duct tape is the "speed tape" is slightly stronger and handles temperature differences better. It still comes down holding the plane together with tape which doesn't exactly fill me with confidence. Especially if it's a Boeing plane.

1 Like

tj-Merry Mischief Maker

Dec 17, 2024 at 6:14pm
In response to "Need for Speed"

I suppose they call it speed tape to keep people calm. If they called it what it is, temporary repair tape, they may lose passengers.

2 Like

Waltz Invictus

Dec 17, 2024 at 9:43am
In response to "Need for Speed"

Ichabod Crane-- Mince Pie

, I've heard that speed tape is extremely expensive. I can only assume the maintenance guys at Delta got a five-finger discount. But if you're going to steal from work, might as well go big and not just lift a box of paper clips.

1 Like

Ichabod Crane-- Mince Pie

Dec 17, 2024 at 9:06am
In response to "Need for Speed"

One of the last jobs I had was with Delta. I did ask the maintanace guys about that tape once and they pointed out high speed will rip duct tape or any other tape right off. They used it personally for their boats sometimes because of its strength. But looking out a window at 400 miles an hour and 30 thousand feet up and see duct tape on the wing. Certainly made me nervous first time I saw it.

3 Like

August 6, 2024 at 10:37am

#1074837

Hall of Fame

The article I'm sharing today, from SciAm, is a bit esoteric. To me, it illustrates why common sense is neither, and why probability can get really weird.

See Why Everyone Gets the Monty Hall Puzzle Wrong
How to finally wrap your mind around the uniquely counterintuitive Monty Hall dilemma

In high school statistics class, my teacher presented a probability problem that haunts me to this day. It was a puzzle inspired by the TV game show Let’s Make a Deal and named after its longtime host, the late Monty Hall.

Several generations of people would never have heard of this guy if it weren't for this puzzle. And it's not like he came up with it; he's just part of the setup. As for the details of the puzzle itself, the article provides a fun illustration of it, but, in short:

There are three doors in front of you.
Behind one of the doors is a new car. Behind the other two doors are goats.
The host invites you to pick a door, any door.
The host throws open one of the doors you did not select, revealing a goat.
Then the host gives you the option of switching your [initial selecton].

Let's leave aside for the moment the relative value of cars and goats. I don't want a new car. I'd be responsible for taxes and insurance on the thing, though I could maybe just sell it for money. But right now, I could really use a goat; you wouldn't believe the condition of my back yard.

Most people think it doesn’t matter whether they stick with their original choice or switch to the other unopened door because the odds are 50–50—that it’s nothing more than a coin toss.

Yeah, that's just common sense. (Which is why I hate "common sense.")

But you should always switch doors. You win two thirds of the time if you switch and one third of the time if you stay. In other words, switching doors doubles your chance of winning.

I have a quibble about "doubles your chance of winning:" If you buy two lottery tickets instead of one (assuming different numbers), that doubles your chance of winning from, I don't know, 1 in a billion to 1 in half a billion. Wow. But, okay, in this particular formulation, it's meaningful.

This counterintuitive problem, first described in relation to the TV show in 1975, is taught in introductory math and statistics classes across the world. But it was widely popularized in 1990 in Parade magazine. After writer Marilyn vos Savant wrote about the puzzle in her Ask Marilyn column, she received an estimated 10,000 furious letters declaring the answer she gave was wrong, including 1,000 or so that were signed by people with a Ph.D. in their title.

It's entirely possible to be smart and wrong. Even if you have a Ph.D. Even if your name is Marilyn vos Savant.

How are the odds not 50–50? Since that fateful column, mathematicians, psychologists and philosophers have been trying to understand what makes this answer uniquely hard to grasp. They’ve found that some of the most common cognitive biases may be to blame, along with a core misunderstanding of how probability works.

The article, of course, goes into the details, and I won't rehash them here. But I will point out another glaring problem we often have with probability, which is almost always illustrated for me when I'm at a blackjack table.

Put as simply as I can, you have the choice in blackjack to stand (keep the cards you have) or hit (dealer gives you another card). The actual probabilities vary, of course, and there can be other complications, but in most cases, you want to either stand or hit, and one of those two has better odds of not losing the game.

But a probability is just that; it's not a certainty. Even if it pays off most of the time (with "most" being anywhere from 50.1% to 99.9% or thereabouts), it won't pay off all the time.

In blackjack, if your strategy fails, you just pony up another bet and play again (or give up in disgust and visit the buffet). But the Monty Hall problem is a one-off. You'll never again be faced with maybe getting a shiny new car or maybe getting a poison-ivy-eating goat, so you only get that one shot. And that one shot is, according to the math, about a 2/3 chance: better than 50-50, but in the end, you might still "lose."

My point being that even the best strategies can fail, and sometimes, we have a problem confusing "probably" with "certainly." And that one data point doesn't mean anything; if you lost on an 80% probability win, that doesn't mean that the next one doesn't have an 80% probability.

Which is why I quit giving probability lessons to other blackjack players.

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