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 14 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 21 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
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

Dec 23, 2024 at 4:55pm
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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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

May 6, 2024 at 9:44am

#1070601

Brain Farts

I think my random number generator is determined to clear out all the Cracked links from my queue.

5 Deaths That Make You Say, ‘Well, He Was Asking For It’

That's a comedy site, and death is very much not funny... except when it is, like when someone falls off a cliff trying to take the perfect selfie.

When someone brings about their own downfall, people sometimes say they were hoisted by their own petard. We avoid using that expression ourselves because what exactly is a “petard” anyway?

Now, that's French I knew before I started learning French. The first translation I heard was "little fart." The linked article goes into more detail. Not about farts, but the word "petard."

5. The YouTube Demonstration of How Bullets Can’t Go Through Books

We like to make fun of those damn kids and their tikkytokky, but people posted Stupid Human Tricks on YouTube since approximately the day it was released.

This particular one, though, is from 2017, well over 10 years after the site launched.

As for Ruiz, he will sadly never get to fulfill the promise he made with some of his last words: “Every week, I’m gonna be bringing you guys new videos!”

This is why I never make promises. As soon as I start to do so, my mind goes to all the dumb ways I could die and thus break the promise. I'd rather die than break a promise, so I avoid having to make that choice.

4. The Thieves Who Stole Cobalt-60

You know, this wouldn't be the karmic justice that it obviously is if it had happened back when Marie Curie was doing her Nobel Prize-winning shit. But this was in 1999, long after Chernobyl, and in another part of the former USSR.

One of the thieves dropped dead within 30 minutes. This was the member of the gang who actually held the material. Sources differ on exactly what happened to the rest, with some saying that two others died next. At least one thief did survive, which meant he went to the hospital and then got arrested.

And sadly, not sentenced to help clean up Chernobyl.

3. Ripping Down a Crash Mat, and Crashing

In case it wasn’t obvious, all the men had been drinking.

Don't freakin' blame booze for this. Booze enhances what's already there, be it stupidity, like these guys, or, in my case, genius.

2. The Trophy Hunter, Crushed by His Trophy

Every so often, you’ll hear about some rich American going to Africa to kill a big-game animal and pose with the kill. Theunis Botha was not one of these casual trophy hunters. Botha was an expert trophy hunter, who led hunts in Zimbabwe and South Africa for decades.

This one is satisfying on at least two levels. Not only do we get treated to jungle justice, but also we can make "Many Bothans died" Star Wars reference jokes.

1. Joseph Stalin, for Being Stalin

One of the few good things you can say about Adolf Hitler is that at least he killed Hitler.

Apparently, one can say something similar about Stalin, only his victim was Stalin. It was just a little more indirect, in his case. Kind of a Rube Goldberg self-destruction machine.

His staff did not summon his personal physician. They couldn’t, as the man was currently being tortured in the basement of border security headquarters.

There's a lot more fatal idiocy you can read about at the link that contributed to his death, but ultimately... he was hoist by his own petard.

3 Like • Comment

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