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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 20 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🎁

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

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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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

August 12, 2024 at 11:32am

#1075125

Lost Soles

Maybe you think you already know the answer to this headline question. Maybe you're even right.

Why Do People Toss Shoes Over Power Lines?
‘Shoefiti’ is everywhere—but not everyone agrees on what it means. Some suspect it's innocent, while others ascribe darker meanings to a dangling pair of kicks.

Odd as it may seem, power lines have become a somewhat popular source of urban fascination.

That's because the power companies are too cheap to bury the lines.

People have wondered why they sometimes sport brightly colored balls...

I always figured it was so the line locations would be visible to pilots.

...why chunks of trees sometimes hang from them...

Because a tree broke and happened to get caught on a power line?

...or why birds love to use them as perches.

Who the hell knows what goes on in a bird's brain? But I'd guess "visibility."

I could click on all the links they provided, but I didn't.

Why do people often see pairs of shoes dangling from power lines?

Because they happen to be looking up at the time?

Lauren Cahn of Reader’s Digest covered a few possible reasons, and not all of them are benign.

The first urban legend I heard was definitely not benign: that it was a gang killing memorial. I've also heard it's a prostitute signal (like a bat-signal, but for whores), or that it indicates a drug dealer's location.

One time, driving through the outskirts of Seattle, I saw what must have been two dozen sneakers, each pair laced together, clustered on a power line like smelly, dingy grapes. I just figured a gang killed someone as he was buying drugs while banging a prostitute.

One popular theory holds that the shoes may be a signal that there’s gang activity in a given neighborhood... Plausible? Sure. But Cahn couldn’t find any police departments that would confirm.

That's because police are basically another gang.

Other stories echo the idea that the shoes could be an impromptu memorial.

Gosh, if you repeat something often enough, it must be true.

Other sources cited nothing more than juvenile mischief.

Sure, but no one wants to believe something that innocent.

The most innocuous explanation? That it’s simply a rite of passage. One columnist for Hidden City Philadelphia wrote in 2012 that the practice was common in the 1970s as a way of discarding old or outgrown sneakers.

Getting a pair of tied-together sneaks to hit an overhead line in such a way as to snag them on it takes skill and/or patience. Or so I'd expect.

More recently, students at the University of Michigan observed that the act was simply commemorating their graduation.

What, they don't let 'em throw mortarboards into the air?

There’s likely no one motive for the shoe-tossing. It may, however, be in decline.

Given the price of sneakers, I'd bet people are wearing them until they fall all the way apart. Also, aren't a lot of them Velcro now?

Whatever the motive for tossing them, the shoes pose a risk of interrupting the power line's performance.

I suppose if there are multiple lines and the shoes touch both and they're not rubber-soled and they get wet...

My best guess? Different reasons in different regions. It's not like all the shady characters in the world got together on the internet and said, "Okay, from now on, shoes thrown over power lines means "drug dealer within one block."

I mean, they could. But that's just asking to get them monitored by a better-funded gang.

4 Like • Comment • View 1 comment

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