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Wednesday, Apr. 24, 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

about 7 hours 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?

1 Like

Lazy Writer est 4/24/2008

about 10 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.

1 Like

Aennaytte: Free & Wild in GoT

about 10 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.

1 Like

Aennaytte: Free & Wild in GoT

about 10 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.

1 Like

Aennaytte: Free & Wild in GoT

about 11 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 ...

1 Like

Aennaytte: Free & Wild in GoT

about 11 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 11 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

May 7, 2022 at 12:21am

#1032067

Dien Bien Phu

Thought I'd experiment with a new, occasional format today. I learn something, maybe you learn something.

On This Day
7-May-1954
End of the Battle of Dien Bien Phu

I have an appreciation for history, sure, but one aspect of it that never really grabbed my attention is detailed war chronicling. War is usually too depressing, even for me, so normally I content myself with having a vague idea of the timeline, major players, and general outcome.

And even there, some military actions are more on my radar than others. The big US conflicts, of course; the Sino-Japanese wars, the Seven Days War in the Middle East, that time the UK took back the Falklands and... ummm... well, apart from the really historical ones like the time Hannibal crossed the Alps on elephants (come on, just picture that), I'm pretty ignorant.

I was almost completely ignorant about the Battle of Dien Bien Phu.

I mean, I had a vague notion that the French got their asses handed to them in Vietnam about a decade to two before we went in and got our asses handed to us, but that's about it. I wonder if this particular battle is where they got their offensive and undeserved reputation as surrender monkeys? Look, everyone's lost a war. Including us. Several times to Canada, for fuck's sake.

I'm not going to rehash most of what's already at that link in the title, just make a few of my own observations, as is my wont:

The United States was officially not a party to the war, but it was secretly involved by providing financial and material aid to the French Union, which included CIA contracted American personnel participating in the battle.

I doubt this was the first time we were secretly egging on a war, nor was it the last. The Cold War wasn't, always.

The Peoples Republic of China and the Soviet Union similarly provided vital support to the Viet Minh, including most of their artillery and ammunition.

Yay, proxy wars!

By 1953, the First Indochina War was not going well for France.

This being Wiki and not Cracked, and having little historical perspective of the overall conflict, I'm not sure if this is comic understatement or not. I'm going with yes.

Most of the page deals with the actual weeks-long battle, and is of value, but my purpose here is to a) snark on things and b) focus on this particular day in history, the day the battle officially ended.

But the "Battle" section is interesting mostly because of the names the French gave their fortifications. French women's names. It's just so... very... French.

So, on May 7, 1954, they record this message:

The last radio transmission from the French headquarters reported that enemy troops were directly outside the headquarters bunker and that all the positions had been overrun. The radio operator in his last words stated: "The enemy has overrun us. We are blowing up everything. Vive la France!"

Bad. Ass. Even in defeat.

Afterwards:

Public opinion in France registered shock that a guerilla army had defeated a major European power.

Yeah. History doesn't repeat itself. It echoes.

One final note, from the page:

The French government in Paris then resigned, and the new Prime Minister, the left-of-centre Pierre Mendès France, supported French withdrawal from Indochina.

Pierre Mendès France. For whatever reason, I'd never heard of him. But that name. It would be like if we elected a President named Rock America. If that President was from a liberal Brazilian Jewish family.

Which, you know... we could do worse.

4 Like • Comment • View 1 comment

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