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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 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.

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