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.

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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

June 27, 2024 at 9:37am

#1073235

Wave Rider

Art History. Another subject I know almost nothing about (though, no, I'm not here to mock liberal arts degrees). Lack of knowledge has never stopped me from blogging about something, however. So, from The Conversation way back in 2017:

Friday essay: from the Great Wave to Starry Night, how a blue pigment changed the world

If you're wondering why I picked this article in the first place, well, there are several reasons, not least of which is that it's tangentially relevant to the title of my blog.

How? Well, there's a detail from Great Wave there at the article link. Now, look at this detail from the fractal Mandelbrot Set,

an iconic image of the strange and wonderful properties of complex numbers.

This is either coincidence, or Hokusai was tapping in to the same chaotic properties of nature to create that painting.

I'll save the other big reason for the end.

Hokusai’s The great wave off Kanagawa remains the enduring image of Japanese art.

And it even includes Fuji-san, which as I understand it is at least as important to Japanese art as tentacles are.

Although diminutive in scale, the importance of Hokusai’s “Great Wave” cannot be overstated. The work profoundly motivated the French Impressionist movement, which in-turn shaped the course of European Modernism, the artistic and philosophical movement that would define the early 20th century.

I'm going to have to take the author's word for that, because, like I said, I don't know much about art history.

But I do detect a whiff of Euro-centrism there. I could be wrong about that, though.

The most immediate and attractive aspect of Hokusai’s wave is its colour.

I mean, maybe? Again, I'm more interested in the fractal boundaries, its depiction of the froth on the waves. But there's good reason I'm not an art person.

The story of this blue pigment highlights the role of cultural exchange at the heart of creative discovery and ranks among the more contradictory tales in the history of art. The vibrant hue, long considered to be quintessentially Japanese, was actually a European innovation.

Okay, that's more than a whiff of Euro-centrism.

I'm not saying it's wrong, mind you. But lots of Japanese things are quintessentially Japanese not because they necessarily invented it, but because they elevated it. Tea, for example. Automobiles. Whiskey. Tempura. Anime.

In truth, it had been invented half a world away, 130 years before Hokusai’s wave broke, in an accident involving one of Europe’s most colourful figures: Johann Conrad Dippel. Born in the actual “Castle Frankenstein” in Germany in 1673, the enigmatic theologian and passionate dissector believed the souls of the living could be funnelled from one corpse to another, thus becoming the rumoured inspiration for Mary Shelley’s masterpiece, Frankenstein.

"most colourful figures?" Oh ho! I see what you did there!

In his thirties, Dippel had become captivated by the proto-science of alchemy, but like so many in the profession, had failed to convert base metals into gold.

If the spelling of "colour" hadn't tipped me off already, this sentence would peg the author as very British. Because that's a fine example of the British art of understatement. No alchemist was able to convert base metals into gold; any that said they did were perpetrating hoaxes.

He instead settled on the apparently easier task of inventing an elixir of immortality. The consequence was Dippel’s oil, a compound so toxic that two centuries later it would be deployed as a chemical weapon in World War II.

Also a quintessentially British sense of irony.

Well, the author is actually, apparently, Australian. But they're even better at irony and understatement.

Anyway, the article goes into the history of blue. Well, kind of. It may seem odd to us today, when we can go to Blowe's and get house paint in any one of millions of shades, but back in the old days, certain colors were precious, rare, or even unattainable. There's a reason purple was associated with royalty in the West.

Then:

Perhaps the single most vividly identifiable influence upon the European modernist founders is Van Gogh’s celebrated Starry Night, which owes everything to Hokusai’s blue wave from its colour to the shape of its sky. In letters to his brother, Van Gogh professed the Japanese master had left a deep emotional impact on him.

As regular readers might know, I'm fascinated by cultural exchanges like this. I don't have to know anything about art history to follow what the article is saying.

In reality, Hokusai had skillfully blended European colour and structure with Japanese motifs and techniques into a seamless work of international appeal.

One of my most prized possessions is a T-shirt that a friend gave me. It features a copy of the Great Wave. But towering over the Wave, the doomed boat, and Fuji-san is one of the other great contributions of Japan to world art and culture:

Godzilla.

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