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:

Annette in France

about 11 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

about 15 hours ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

about 15 hours ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

about 20 hours ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

about 21 hours ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

about 21 hours ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

a day ago
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

2 Like

Cubboo!

a day ago
In response to "Get Baked"

Yes! I discovered this a few years back (slitting the potato about 1/4 - 1/2" lengthwise, then one or two slits across the longer slit.) It came out wonderful! It seems like this was called an English jacket potato, or something like that.

2 Like

Annette in France

a day ago
In response to "Get Baked"

I'll try your oven baked potato in winter when it's sensible to run the oven for 90 minutes.

2 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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