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:

Wordsmitty ✍️

about 2 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

about 24 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 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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