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:

Write_Mikey_Write!

about 8 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 10 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

1 Like

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').

1 Like

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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

February 17, 2022 at 7:37am

#1026872

Over the Rainbow

Last day of being away from home... until next week.

Even Rainbows Have a Dark Side
Humans were once united by the belief that you should never, ever point at one.

I find that extremely difficult to believe. Not because it's a silly superstition, but the idea that "humans were once united by" anything.

Both had learned as children that if you broke the taboo, your finger would become bent like a rainbow.

Right, and if you keep making faces, your face will freeze like that; if you crack your knuckles you'll get arthritis (to be fair, you might, but not from cracking); and if you watch too much porn, you'll go blind.

*looks at medical bill for cataract surgery*

...okay, so that last one might be true.

The first hint that he was on to something bigger, something truly boggling, came from a report of a prohibition on pointing to rainbows in India. “It did not take long for the shock to set in,” he wrote. The report suggested “the rainbow taboo”—as he would come to call the phenomenon—was not confined to Southeast Asia.

Oh, it's in Southeast Asia and India? Wow, it must be all over the world.

He would soon amass evidence for the rainbow taboo—in some form or another—in 124 cultures. The prohibition turned up in North America, among the Atsugewi of northern California and the Lakota of the northern plains; in remote parts of Australia and isolated islands in Melanesia; among the Nyabwa of Ivory Coast and the Kaiwá of Brazil.

Okay, okay, jeez, I get the point. You make one joke...

The belief was not found in every culture, according to Blust’s search, but it was present globally, across all inhabited regions.

See? I told you we weren't united.

Less commonly—such as in parts of New Guinea and Australia—the ill effects would befall your mother.

When I was a kid and my mother made me do something I didn't like, or scolded me for doing something I did like, I'd deliberately stomp on the cracks in the sidewalk.

I was an asshole kid.

This, by the way, took significant effort and patience on my part, because where I spent my childhood there were no sidewalks for miles, so I had to wait until someone let me ride with them into town. If it was my mom, I'd leap onto a crack, look back at her, jump to another one, look back, stomp, look... getting more and more disappointed each time (never mind that if I'd been successful, no one would have been able to drive me back home).

As her back never broke, I lost all faith in kid lore. But later, when I learned about how concrete (such as that in sidewalks) works, I discovered that as concrete cures, it develops microfractures throughout its matrix. You can't see them (hence, "micro") but if you're walking on a sidewalk you cannot avoid stepping on dozens of cracks with each and every pace.

The relative lack of mother's back breakage in the world should be enough to put people off such taboos, but apparently not.

A final recurring idea was that, should you accidentally point to a rainbow, there were remedies. You could wet the offending digit; or put it into a bodily cavity like your mouth, anus, or belly button; or, according to the Javanese version of the taboo, plunge it into a pile of buffalo dung.

Or any of the above, in random order?

Anyway. No, the article isn't about fact-checking the taboo; it's about the very interesting widespread nature of the belief and speculation about the reasons for it. And while even that didn't unite humanity as one shining, glorious people singing under the rainbow, it's a fascinating glimpse into peoples' beliefs.

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