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 17 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 20 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 11, 2024 at 11:46am

#1070913

Crank

Damn, people in the past sure had barbaric ideas about things like medicine... and punishment.

Victorian Crank Machine: a soul-crushing gadget designed to break prison inmates

I'm hereby calling dibs on Victorian Crank Machine for my steampunk band.

In the grim confines of 19th century British prisons, wardens sought ever more sadistic ways to punish inmates through forced labor.

Gosh, this never happens today.

One diabolical device they landed on was the crank machine.

Another was the treadmill,

which you've probably used voluntarily at least once in your life.

The crank was a hand-powered rotary mechanism that served no purpose other than to torment prisoners. Locked in isolated cells, inmates had to turn a crank handle connected to paddles that churned sand inside a box. The only output was soul-crushing tedium.

If this happened in the US, I'd blame someone taking seriously the specious claim that "idle hands do the Devil's work." I don't know; maybe enough religious nutbars stayed in the UK for it to be true there, too.

Also... let's pretend for a moment that it did serve some purpose, like, I don't know, turning wheat into flour or running a coal conveyor or the urchin grinder, or whatever else the Victorians needed done. Would that have made it somehow better?

To earn basic sustenance like food and water, prisoners were expected to complete a 10,000-14,000 revolutions per day on the crank, often working late into the night by candlelight if they fell behind quota.

I guess this was before the lightbulb was invented, or they could have used the power to keep the bulb lit. Maybe; I didn't do the math. But regardless, why would someone need light to do this task?

Mercifully, the crank machine faded from use by the early 20th century as more humane concepts of rehabilitation took hold.

I suspect the phrase "more humane" is relative. One wonders what punishments of today will be looked back on in a century with great disgust.

1 Like • Comment

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