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Wednesday, Sep. 11, 2024

10:22pm EDT

Members: 937

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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:

Aiva Raine

about 7 hours ago
In response to "Drop and Gimme Twenty"

Happy 20th. We're glad you joined and doubly glad you've stayed this long. I look forward to reading your daily blog as part of my own daily ritual.

1 Like

Wordsmitty ✍️

about 9 hours ago
In response to "Drop and Gimme Twenty"

Wow. That sounds mature for twenty. Next year you can get some booze to celebrate.

I'm sure we're all glad you did sign up. Enjoy the day.

1 Like

Annette aka Din Djarin

about 9 hours ago
In response to "Drop and Gimme Twenty"

Happy 20th to you!

My username was octobersun2 for the same reason. My first attempt to join somehow didn't work.

1 Like

🌕 HuntersMoon

about 10 hours ago
In response to "Drop and Gimme Twenty"

Happy Anniversary, you old dog...

1 Like

Kåre Enga in Montana

Sep 9, 2024 at 10:57am
In response to "Time Out"

"As long as everyone agrees to it" is paramount.

The Polynesians navigated by the stars. No clocks needed. A different approach. Still... all relative.

1 Like

Legerdemain

Sep 9, 2024 at 10:47am
In response to "Time Out"

I'm not scientific, at all. But what about Earth's wobble? Who accounts for that?

1 Like

🌕 HuntersMoon

Sep 9, 2024 at 10:20am
In response to "Time Out"

This was so heavy... I think I lost 26 microseconds reading it.

1 Like

Elisa: Middle Aged Stik

Sep 8, 2024 at 12:04pm
In response to "Writing It Off"

I am late to the party with my comments, but here we go.

Used to be, I wanted to get published.

I used to want to be published in terms of stories. Nowadays I'd rather be published in terms of photos. And self publishing may be my only option in that regard.

but the travel writing field is crowded with young influenzas who can't write for shit, but look good in selfies.

And before that it was crowded with middle aged socialites with nothing better to do (but did have some level of writing skill).

Other people here have dealt with the limit by deleting old entries. I won't do that. Most never have to worry about the limit, and that's perfectly okay.

I admit that as of late I don't let my book items hit the entry limit. Part of it is my entries are super long (current average being *checks notes* 7.48 kb in size). Part of it is I also feel like book items should end after at most five years. I think of my old paper diaries and how they felt more meaningful when I didn't let their use drag on forever. If I stayed consistent, they'd end maybe at most after a year and a half. They often wrapped up sooner than that. Even moving to online journaling, I appreciate a journal that is finite not based on the number of entries but by time elapsed since the book started having entries added to it. My current journal may last a tad longer than my previous ones, but I imagine I'll wrap it up in late 2027, maybe early 2028. We'll see how much space I have to work with. In the old days, 1000 entries was considered a major achievement of an end point. I hope I can get that much out of it, as I think that's more reasonable. I do think 3000 entries of my kind of work in one place is a bit much.

1 Like

🌕 HuntersMoon

Sep 8, 2024 at 11:00am
In response to "Revisited: "Orange You Glad This Wasn't You?""

Maybe this would help... or at least attract your attention?

1 Like

Robert Waltz

Sep 7, 2024 at 11:39pm
In response to "Spit in the Ocean"

Both commenters today had already received MBs from me this week, so I went back to every entry this week and picked a comment at random.

The final Birthday Week MB goes to:

🌕 HuntersMoon

Thanks, everyone, for reading and participating!

1 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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