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 an hour 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 3 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.

May 27, 2023 at 11:37am

#1050145

Ur Gent

Writing—the concept, not this website—has been around for at least six thousand years. It's evolved somewhat in that time.

Marvel meets Mesopotamia: how modern comics preserve ancient myths

It's widely known, thanks to movies and TV shows, that some of Marvel's characters are its take on Norse gods. This article delves a bit deeper into the past than the Asgardian characters.

Ancient Mesopotamia, the region roughly encompassing modern-day Iraq, Kuwait and parts of Syria, Iran and Turkey, gave us what we could consider some of the earliest known literary “superheroes”.

It also gave us beer. Coincidence? I don't think so.

One was the hero Lugalbanda, whose kindness to animals resulted in the gift of super speed, perhaps making him the literary great-grandparent of the comic hero The Flash.

I want to reiterate here that I interpret the words "literary" and "literature" in their broadest sense; hence, comic books are literature, while movies and TV are not. The difference, to me, is whether one reads the words, or listens to actors or storytellers speaking them (even if you tend to watch shows with subtitles, as I do). This isn't an attempt at a prescriptive definition, but the point of view I take when discussing these things.

Many ancient stories probably started out in an oral tradition, and only became literature when they were written down.

But unlike the classical heroes (Theseus, Herakles, and Egyptian deities such as Horus), which have continued to be important cultural symbols in modern pop culture, Mesopotamian deities have largely fallen into obscurity.

Except, if you're me, the ones involved with beer. And D&D.

An exception to this is the representation of Mesopotamian culture in science fiction, fantasy, and especially comics. Marvel and DC comics have added Mesopotamian deities, such as Inanna, goddess of love, Netherworld deities Nergal and Ereshkigal, and Gilgamesh, the heroic king of the city of Uruk.

Many writers delve into the past for inspiration; comics writers are no exception. Like when someone retells a Shakespeare play, as with West Side Story.

The Marvel comic book hero of Gilgamesh was created by Jack Kirby, although the character has been employed by numerous authors, notably Roy Thomas. Gilgamesh the superhero is a member of the Avengers, Marvel comics’ fictional team of superheroes now the subject of a major movie franchise, including Captain America, Thor, and the Hulk. His character has a close connection with Captain America, who assists Gilgamesh in numerous battles.

And no, he hasn't shown up in the MCU yet. Unless I've missed an Easter egg somewhere, which is always possible.

Gilgamesh’s first appearance as an Avenger was in 1989 in the comic series Avengers 1, issue #300, Inferno Squared. In the comic, Gilgamesh is known, rather aptly, as the “Forgotten One”. The “forgetting” of Gilgamesh the hero is also referenced in his first appearance in Marvel comics in 1976, where the character Sprite remarks that the hero “lives like an ancient myth, no longer remembered”.

That's recent, as comic book characters go. The genre is generally considered to begin with Superman in 1939.

Unlike many ancient tales and myths, I knew about Gilgamesh before the comics version. One should never assume that a comics version is a reliable adaptation of ancient stories.

Story-telling has been recognised since ancient times as a powerful tool for imparting wisdom. Myths teach empathy and the ability to consider problems from different perspectives.

I would argue that this is a primary purpose of storytelling, ancient and modern. Not limited to myths.

A recent study has shown that packaging stories in comics makes them more memorable, a finding with particular significance for preserving Mesopotamia’s cultural heritage.

While a single unreferenced study is no grounds for drawing conclusions, this tracks for me.

The myth literacy of science fiction and fantasy audiences allows for the representation in these works of more obscure ancient figures. Marvel comics see virtually the entire pantheons of Greece, Rome, and Asgard represented. But beyond these more familiar ancient worlds, Marvel has also featured deities of the Mayan, Hawaiian, Celtic religions, and Australian Aboriginal divinities, and many others.

That's because those of us who read science fiction and fantasy are, in general, more knowledgeable and intelligent (not to mention better looking) than those who don't.

Again, though, be wary of taking modern interpretations of these figures, whether in comics or RPGs, as definitive.

In the comic multiverse, an appreciation of storytelling bridges a cultural gap of 4,000 years, making old stories new again, and hopefully preserving them for the future.

Which leads me to my main purpose in featuring this article: the idea of storytelling itself.

Before writing, stories were passed down orally (though possibly with props, like an early form of theatre), and they would have changed with changing technologies and societies. Once you write something down, though, it's preserved, like a mosquito in amber. It becomes harder to interpret after, for example, war wipes out a neighbor that was mentioned in the story, or someone invents the axle or whatever, making the story less relevant. Just as someone today might view a movie from the 70s and note that a particular predicament might have been easily solved had mobile phones been a thing then.

So, in my view, these adaptations are necessary for preservation, breathing life into old tales.

Thus, we exist in a time when we can have the best of both worlds: the original, preserved; and the reinterpretation.

It could be—and has been—argued that today's comic book superheroes are the modern take on ancient mythology in general, what with their focus on exceptional abilities and superpowers. You get that a bit in other media, but nowhere is it more obvious than in the realm of graphic storytelling.

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