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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:

Elisa the Bunny Stik

about 15 hours 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

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:04pm
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 ...

1 Like

August 12, 2020 at 12:02am

#990565

I Got a Rock

When the answer to the question, "Have you been living under a rock?" is "Yes."

https://www.cracked.com/article_28076_a-brief-history-giant-rock-americas-most-b...

A Brief History of Giant Rock, America's Most Bonkers Landmark

Americans are not known for their creativity in naming landmarks.

It's called Giant Rock, because desert living requires many things, but wild flights of poetic fancy aren't one of them.

Nor are umbrellas.

There's certainly nothing remarkable nearby, except for the time machine.

Way to bury the lede.

Over the years, it's also been a family home, a center of pilgrimage, the scene of a bloody standoff, a potential rival to Las Vegas, and the galactic antenna for the wisdom of interdimensional aliens.

Pretty sure I saw this episode of Star Trek.

The giant rock's modern history starts in the 1930s, when a German guy named Frank Critzer stumbled on it, conveniently located in a part of the Mojave Desert so remote the scorpions had to get two buses and a cab just to murder you.

Now, now, surely that is hyperbole. Scorpions have their own custom means of transportation. Pretty sure it involves teleportation gates in shoes.

Besides, Critzer was reportedly quite fond of waving a shotgun at any distant neighbors who came snooping around. Since ornery old shotgun weirdos are a crucial part of the desert ecosystem, nobody asked too many questions.

This part rings true.

Using only dynamite, pickaxes, and insanity, he managed to excavate a large cave underneath the rock...

You can accomplish quite a bit with those tools alone.

Before long, he was welcoming over a plane a day, and fantasizing about converting the whole area into a Vegas-style winter resort.

Oh, the missed opportunities.

These plans were destroyed by the arrival of WWII, when the other scattered inhabitants of the desert reported seeing mysterious flares in the mountains, as well as missing dynamite.

I have to think that at least one person made the obvious connection, but this is the Mojave.

Public opinion turned against the mysterious German with the massive radio antenna on top of his rock, particularly after the FBI swept up the Silver Shirts, a group of Nazi sympathizers attempting to convert a Hollywood mansion into a self-sustaining headquarters for American fascism.

Eighty years too early and on the wrong coast.

The group went down into the house under the rock, where Critzer exploded. Which probably answered their questions about the dynamite.

"Oh, there it was."

In any case, Critzer was instantly converted into "wallpaper," while all three deputies were badly wounded. One just about managed to drive 40 miles to the nearest telephone to get help.

This should be a movie. This needs to be a movie.

The house under the rock was left abandoned until the arrival of George van Tassel, who worked as an aircraft engineer for Howard Hughes and claimed to have met Critzer back when he first moved to California, although his story of the meeting might be the most implausible thing in this whole article.

That's... that's a high bar to clear.

In 1947, he decided to move his family out there, which involved scrubbing Critzer's dried blood off the walls. Just remember, if your husband ever comes home and says "Honey, we're moving to the desert and make sure to bring your blood sponge!" there's no need to even divorce him, the law says you can just empty the bank accounts and go.

I know better than to take legal advice from Cracked, but this sounds about right.

In 1952, van Tassel announced that his time at Giant Rock had led to him being visited by a race of aliens, who looked like tan white people and spoke in posh English accents.

No mention of a blue police box in the area at the time?

The story goes on to relate how van Tassel next began construction of a time machine called the Integratron, which explains the bit in the first paragraph above.

In 1957, Van Tassel used the rock as a platform to announce his run for president in the 1960 elections.

Thus locking up that all-important scorpion demographic.

But somehow the machine was never finished (in fairness, time machines are complicated, and it would be cynical to bring up the ongoing donation money) and he died in 1978, without ever publishing a complete plan for the machine.

Too bad the original British time travelers took their TARDIS with them.

The internal machinery of the Integratron mysteriously vanished shortly afterward...

Only to reappear suddenly at some point in the future, obviously.

So, yeah, four out of five on TripAdviser, good job George van T!

In some ways, I hope we never cure this kind of insanity. I mean, the human cost aside, stories like this are just too much fun. Well, except for the whole "dynamite wallpaper" bit. That's sad.

Mini-Contest results!

Great answers to my question from yesterday, some serious, one amusing. I thought about giving it to the funny one, in honor of today's amusing article, but the late entry by Words Whirling 'Round is well-written and generally aligns with my own thoughts on the matter. (It's dated August 11, yesterday, so that's not a link to some older item; therefore, I decided it counts.) So the Merit Badge today goes to Words Whirling 'Round. I highly recommend reading the essay; here's another link to it: "Consciousness"

[E].

But as always, I appreciated all the comments and seeing how other people think about the subject of consciousness, and I hope to do this again soon so everyone will have another chance. Probably with something less... mentally weighty.

5 Like • Comment • View 1 comment

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