Complex Numbers (Book)

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Complex Numbers

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:

tj-Merry Mischief Maker

about 6 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

1 Like

🐕GeminiGem🎁

about 10 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

about 12 hours ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 6, 2024 at 9:03am

#1079566

Lo!

Well, that's finally over. Now I just get to grump at holiday season chatter.

Today's article, from BBC, has nothing to do with politics or seasons, and everything to do with the invention you're using right now.

'We were just trying to get it to work': The failure that started the internet

There have been differences of opinion concerning the actual beginning of the internet. It's not like a human birth, or Armstrong's boot on the moon: a clear and obvious transition point. In my view, this article is more about a precursor technology, but a vital one for what the internet became.

On 29 October 1969, two scientists established a connection between computers some 350 miles away and started typing a message. Halfway through, it crashed.

1969 would have been long before "try rebooting and reinstalling all your drivers" would become tech support's second suggestion, after "make sure it's plugged in."

At the height of the Cold War, Charley Kline and Bill Duvall were two bright-eyed engineers on the front lines of one of technology's most ambitious experiments.

I should note, for context, that this was the same year as the aforementioned moon landing. Unlike NASA's stated mission, though, this early attempt at remote networking was in service of more military pursuits.

Funded by the US Department of Defense, the project aimed to create a network that could directly share data without relying on telephone lines. Instead, this system used a method of data delivery called "packet switching" that would later form the basis for the modern internet.

Like I said, not the actual invention of the internet, and military.

It was the first test of a technology that would change almost every facet of human life. But before it could work, you had to log in.

Some things don't change.

But Kline didn't even make it all the way through the word "L-O-G-I-N" before Duvall told him over the phone that his system crashed. Thanks to that error, the first "message" that Kline sent Duvall on that autumn day in 1969 was simply the letters "L-O".

And that's what I find amusing about the story: it's very Biblical. "Lo!" As in "Lo and behold."

On the other talon, I want to think that once they got it working (which, as the article notes, they did, after about an hour), the second message sent over this proto-internet was "Send nudes."

The BBC spoke to Kline and Duvall for the 55th anniversary of the occasion.

The rest is a transcript of that interview. It goes into more depth over what happened (or didn't happen) at the time, but there's no reason to repeat it here.

As compelling as this origin story is, I had this vague memory of a different origin story for the internet, one which took place some years later. In a rare case of me actually looking something up, I found this entry from 2019: "Birthed in Beer"

So if I had to choose which one was the actual invention of the internet, I'd pick the 1970s one, because it involved beer.

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