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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:

🌕 HuntersMoon

about 3 hours ago
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

a day ago
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
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?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

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

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
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.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
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.

1 Like

August 18, 2020 at 12:02am

#991071

As the Turn Worlds (or whatever)

Maybe you have an image in your head, as I did, of a primitive caveman using stone tools to create a stone wheel.

Turns (pun intended) out that the wheel was invented much, much later.

And it's about time for another Merit Badge Mini-Contest! See below.

https://www.wired.com/story/who-invented-wheel-how-did-they-do-it/

Who Invented the Wheel? And How Did They Do It?
The wagon—and the wagon wheel—could not have been put together in stages. Either it works, or it doesn’t. And it enabled humans to spread rapidly into huge parts of the world.

It's not like massive leaps of invention aren't known in modern times, so it shouldn't be surprising that someone made the leap from not-wheel to wheel. What is surprising, at least to me, is how recently (relatively speaking) it happened.

In July 1880, the archaeologist Désiré Charnay discovered the first pre‑Columbian wheel set in the Americas. It was on a small coyote figure mounted on four wheels, and Charnay found it in the tomb of an Aztec child buried south of Mexico City.

It's been widely noted that the Maya, a civilization in Central America, didn't have the wheel. Their other advancements were quite sophisticated -- the calendar, notably. And yet, the civilizations they would have had contact with, such as the Aztecs, did have that invention. I had the opportunity to speak with an actual Mayan once, and his assertion was that the wheel would have made things too easy. My personal theory is that the wheel might have been too sacred to use for mundane purposes; their fabled calendar was represented as a wheel of sorts. Whatever the actual reason, it's lost to time, and I'm not even 100% sure that they never used wheels.

I mention this because, as the article notes, it seems to have been invented on both major continents, independently - not the only time such a thing has happened. The rest of the article focuses on the Indo-European version, which ties in to the entry I made a while back on the origins of most of the languages in Asia and Europe.

The full‑size wagon first appeared approximately 5,400 years ago, and it may be one of the the first inventions in history to go viral. Archaeologists have discovered full‑size carts from southern Iraq to Germany within a few hundred years of each other at a time when cultural barriers were particularly impermeable. The wagon, it seems, was irresistibly useful.

"If you build it, they will come." This is more of a case of "if you build it, you can go to them."

The article is fairly long, so I won't go into it much further. It's there if you want to read it. But I found it interesting because, as I said, thanks to cartoons and whatnot I always thought the wheel was a much older invention. It's one of those things we take for granted now, but as noted at the link, there's really not anything in nature that could have inspired it, not like, say, you can watch birds and go "Wow, I wish I could do that," and then eventually invent flying machines.

Since this article presents the wheel, as used in wagons, as one of the most important inventions of human civilization, I thought I'd make today's mini-contest about inventions. So...

Merit Badge Mini-Contest!

Other than the wheel, what's the most important invention, or innovation, that we've made, and why?

The response I like best gets its author a Merit Badge that I think is appropriate. As usual, you have until midnight WDC time (Today is August 18.) Just comment here below. I have a few in mind, but I'd rather hear from others. You can be serious or funny; I can certainly go either way. And there's no limit on how far back you want to go; prehistoric or modern will do.

3 Like • Comment • View 8 comments

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