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 9 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day 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.

2 Like

🐕GeminiGem🎁

a day 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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day 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

September 17, 2024 at 10:00am

#1076900

Uncovered

My random number generator is working just fine. Randomness just sometimes leads to clustering; hence, another one from Cracked today.

5 Secrets Found Buried at Famous U.S. Landmarks
Until we opened a secret panel, no one realize this planned for the apocalypse

No, this article isn't a stealth ad for some new Dan Brown novel.

This Fourth of July, we want you to celebrate America by visiting one of the nation’s most famous sites. We also want you to bring a shovel with you, so you can dig for treasure.

Yes, this article has been in my queue for over two months. It happens. There are older ones. It's mathematically impossible for me to get to all of them before I leave for Europe, and I won't be doing this format there, so they'll be even older when I get back.

5 The Lost Town at the Hoover Dam

The whole point of most dams is to flood a portion of the upstream river. As it is a river, chances are people lived next to it. Some of those people lived in towns. So it shouldn't be surprising that "lost towns" exist under dam lakes.

With Hoover Dam, that reservoir is Lake Mead, and when the country flooded the area to create Lake Mead in the 1930s, there happened to be a town in a way. Fortunately, it was a town full of Mormons, and Mormons are great at moving around, so it was a simple matter to pay them all to go elsewhere.

Pay? good thing they weren't Utes or Paiutes. Because they wouldn't pay Utes.

That joke is not funny.

Of course, if you’re not interested in old Mormon buildings, there’s also the chance that you’ll stumble upon one of the several skeletons that have turned up as Lake Mead has dried up. No, there is no straightforward explanation for how those bodies got there, and investigations are still ongoing.

That's sarcasm, right? Because there's a desert city nearby that was famously run by the Mob for decades.

4 The Other Hollywood Sign

The Hollywood Sign wasn’t originally a landmark just designed to sit in the background as people walk through Hollywood.

It also serves as the only universally recognizable structure for establishment shots of Los Angeles. The area doesn't exactly lend itself to massive, easily-recognizable structures like the Empire State Building or the Eiffel Tower. Blame Andreas; it's his fault.

At the same time as that sign advertised Hollywoodland, a second sign advertised another housing development called “Hillside Homes of Happiness.”

I guess this was before marketing moved out there from NYC.

“Hillside Homes of Happiness” would have been too long a message to erect in 30-foot-tall letters. So, the ad consisted of just the word “Outpost,” since the land around the development was known as “Outpost Estate.” Though these letters were a little shorter than the ones in the Hollywoodland sign, they were lit in red neon, so as to be even more visible.

Okay, now I wish I'd seen that.

3 The Brooklyn Bridge’s Fallout Shelter

Yes, I thought the same thing you're thinking now: the Brooklyn Bridge was built before atomic bombs were a thing.

In 2006, workers tinkering on the Brooklyn Bridge found a vault full of provisions stockpiled by those fearing a nuclear strike. These supplies included drugs, blankets and 352,000 packets of crackers.

This makes no sense. Any nuclear strike would turn all of Lower Manhattan into a glass crater, along with that often-sold bridge.

2 The Radioactive Hazard at the Grand Canyon

In contrast to the other entries here, the Grand Canyon wasn't built by humans.

In 2019, park officials at the Grand Canyon realized that the museum contained three 5-gallons buckets of uranium ore.

Presumably, the museum was.

The uranium’s radiation wasn’t so powerful that the whole museum had an elevated radiation level, but it was powerful enough that the area near the buckets did.

Yeah, I kind of figured that would be the case when they said "ore." It's not like it's refined plutonium.

1 The Capitol’s Forgotten Bathing Area

This final discovery happened in 1936.

No wonder politics stinks. They forgot their bathing area.

If the building would only reopen and expand its baths, and if all legislators agreed to bathe communally, we could reestablish camaraderie and get around to solving everything.

You say that like it's a joke, but it seems to work in Japan.

4 Like • Comment • View 1 comment

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