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Tuesday, Apr. 23, 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/

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Show Recent Comments

10 Most Recent Blog Comments:

GeminiGem of House Lannister

6 minutes ago
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

Lazy Writer est 4/24/2008

about 3 hours ago
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

Aennaytte: Free & Wild in GoT

about 4 hours ago
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

Aennaytte: Free & Wild in GoT

about 4 hours ago
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

Aennaytte: Free & Wild in GoT

about 4 hours ago
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

Aennaytte: Free & Wild in GoT

about 4 hours ago
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

Aennaytte: Free & Wild in GoT

about 4 hours ago
In response to "Going to the Dogs"

I have never been to New York, but it seems that my assessment years ago that Berlin, Germany and New York have some similarities. Both cities are known for being inhabited by rude people. And both places, as I now found out, are dog obsessed. And they have high rents.
Berliners though, have a habit of letting their dogs run off leash. Maybe New York is a little better than Berlin in that regard.

1 Like

Robert Waltz

Apr 21, 2024 at 1:48pm
In response to "Revisited: "Hack This""

Wordsmitty ✍️, far be it from me to speculate on the motivations of space aliens, but why in Xenu's name would they do such a thing to the universe?

1 Like

Wordsmitty ✍️

Apr 21, 2024 at 1:39pm
In response to "Revisited: "Hack This""

I thought humans got bigger brains because extraterrestrials affected our evolution, and we eat processed chicken and eggs.

1 Like

Robert Waltz

Apr 21, 2024 at 10:22am
In response to "Revisited: "Hack This""

🌕 HuntersMoon, ah, yes, my favorite words to hear (or read).

Also, that video is an excellent, if brief, explanation of the conundrum and how to resolve it.

1 Like

September 15, 2021 at 12:03am

#1017439

Uh huh. Sure.

Finally, vindication! This isn't confirmation bias talking at all.

Sarcasm Spurs Creative Thinking
Although snarky comments can cause conflict, a little verbal irony also stimulates new ideas

Yeah, sure it does.

“Sarcasm is the lowest form of wit but the highest form of intelligence,” that connoisseur of witticisms, Oscar Wilde, is said to have remarked.

No, fart jokes are the lowest form of wit.

The reason is simple: sarcasm carries the poisonous sting of contempt, which can hurt others and harm relationships. By its very nature, it invites conflict.

That's only if people detect it. Or, in my case, when I do detect it, I play it straight, like I didn't. That sometimes throws them off balance.

Sarcasm involves constructing or exposing contradictions between intended meanings.

Because no other forms of expression do that.

And yet behavioral scientists Li Huang of INSEAD business school, Adam D. Galinsky of Columbia University and I have found that sarcasm may also offer an unexpected psychological payoff: greater creativity.

Sure, yep. That's why I'm so damn creative.

And sarcasm can be easily misinterpreted, particularly when it is communicated electronically...

All of my attempts at turning Comic Sans into the Official Sarcasm Font have failed miserably.

Anyway, the article goes on to describe a psychology experiment -- like most psychology experiments, the guinea pigs were college students -- to test the author's claim in the headline.

Given the risks, your best bet is to keep conversational zingers limited to those you know well, lest you cause offense.

Nope. I'd rather piss off strangers on the internet. Not here, usually - I have other sites for that.

Anyway, it's an interesting result though, to be serious about it, I have doubts concerning the sample size and lack of diversity in experimental subjects. But it's probably worth pursuing further. Not by me, though. I'll just have to read about it (this article is five years old and for all I know, they've already followed up).

But I'm reminded of an old story -- could be a joke, could be real; I can't tell -- about a college English class where the professor was talking about the double negative. He said something like, "You can string two negatives together to make a positive, but there's no instance of stringing two positives together to make a negative."

To which one student muttered, "Yeah, right."

Mini-Contest Results!

Great comments! Thanks for all of them. Hard to pick just one that stood out, because everyone had some real gems, but I'm going with Kåre Enga in Udon Thani's five rules (though I must note that Lilli's point about tequila is absolutely true). For everyone else, I'll do this again soon! I really want to do MB entries more often; it's just that usually I get caught up in the post and then I forget. I'll try to be better at it, at least 2-3 per month (spaced out so I don't have to worry as much about CR eligibility).

6 Like • Comment

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