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:

Write_Mikey_Write!

about 3 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 5 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.

April 14, 2021 at 12:02am

#1008482

Wagging Tales

Just an interesting tale of tales today.

Fairy Tales Could Be Older Than You Ever Imagined
Jack may have been climbing that beanstalk for more than 5,000 years

Though I'm not sure how solid the foundation for this finding is, it's about storytelling, so it's actually relevant for once.

In a new study published in the journal Royal Society Open Science, a folklorist and anthropologist say that stories like Rumpelstiltskin and Jack and the Beanstalk are much older than originally thought. Instead of dating from the 1500s, the researchers say that some of these classic stories are 4,000 and 5,000 years old, respectively. This contradicts previous speculation that story collectors like the Brothers Grimm were relaying tales that were only a few hundred years old.

Before writing, we're pretty sure people passed down their stories mouth to ear. There's no reason to believe that wouldn't have continued; not everyone in every society was - or is - literate.

It turns out that it’s pretty hard to figure out how old fairy tales are using simple historical data. Since the tales were passed down orally, they can be almost impossible to unwind using a historian or anthropologist’s traditional toolbox. So the team borrowed from biology, instead, using a technique called phylogenetic analysis. Usually, phylogenetic analysis is used to show how organisms evolved. In this case, researchers used strategies created by evolutionary biologists to trace the roots of 275 fairy tales through complex trees of language, population and culture.

While that's interesting and all that, I'd be inclined to want to hear how they justify using phylogenetic analysis for something for which it wasn't intended.

Using the Aarne-Thompson-Uther Classification of Folk Tales, a kind of über index...

More like Uther index, amirite?

...that breaks fairy tales down into groups like “the obstinate wife learns to obey” and “partnership between man and ogre,”...

Both groups being what today we'd call "fantasy."

...the team tracked the presence of the tales in 50 Indo-European language-speaking populations.

It might also be illuminating to use this technique (if it's not found to be fatally flawed) on non-PIE-derived languages. There's a rich culture of fairy tales, or their equivalent, in the East.

But not everyone is certain that the study proves fairy tales are that old. As Chris Samoray writes for Science News, other folklorists are finding fault with the study’s insistence that The Smith and the Devil dates back to the Bronze Age—a time before a word for “metalsmith” is thought to have existed.

Clearly, skepticism is warranted (and encouraged), but in that particular example, so what? The thing about oral traditions is that each generation adds its own spin on the tradition. If environmental, cultural, or (in this case) technological changes occur, they'd probably morph the story to make it more accessible to a younger generation, one that lacks historical context. If you don't believe me, go watch one of the approximately two thousand remakes Hollywood puts out in a year.

Point being, maybe that particular story started out as "The Flint-Knapper and the Evil Spirit." No need for "metalsmith" to have been invented; that would come in after metalworking got going.

So, yeah, I wouldn't take anything here as hard evidence of the age of stories... but we're pretty sure storytelling itself is as old as humanity. Maybe it's when our ancestors started telling stories that we could finally call them "human."

But of course a writer would think that.

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