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:

tj-Merry Mischief Maker

about 6 hours ago
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.

1 Like

🐕GeminiGem🎁

about 10 hours ago
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

about 12 hours ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

about 12 hours ago
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

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 9, 2024 at 10:50am

#1079718

It Was a Dark and Starry Night

When science meets art, are they both elevated? Or do they make each other suck worse? From CNN:

Turbulent skies of Vincent Van Gogh’s ‘The Starry Night’ align with a scientific theory, study finds

I usually groan when I see "scientific theory" in a headline, because the word "theory" means something different in science than it does in everyday speech. In this case, though, my fears proved unfounded.

Which doesn't mean I don't have some issues with the rest of the piece.

Now, a new analysis by physicists based in China and France suggests the artist had a deep, intuitive understanding of the mathematical structure of turbulent flow.

Or, and hear me out here, he was an artist and thus observed turbulent flow in, perhaps, a river or whatever, and incorporated that observation without being able to do math beyond "this paint costs 3 francs and this other one costs 4; which one is cheaper?"

As a common natural phenomenon observed in fluids — moving water, ocean currents, blood flow, billowing storm clouds and plumes of smoke — turbulent flow is chaotic, as larger swirls or eddies, form and break down into smaller ones.

"Chaotic" is another word that means something different in science than it does in everyday speech. Again, though, credit where it's due; the article uses it in the scientific sense:

It may appear random to the casual observer, but turbulence nonetheless follows a cascading pattern that can be studied and, at least partially, explained using mathematical equations.

"Partially" is doing a lot of the work in that sentence.

“The Starry Night” is an oil-on-canvas painting that, the study noted, depicts a view just before sunrise from the east-facing window of the artist’s asylum room at Saint-Rémy-de-Provence in southern France.

There are, I think, a few paintings that even the art-blind (like me) can identify at first glance. Mona Lisa. That Michelangelo thing with God and Human. The Scream. Maybe that one with the farmers. And The Starry Night. But, on the off-chance you have no idea what we're talking about, the article provides helpful illustrations.

Using a digital image of the painting, Huang and his colleagues examined the scale of its 14 main whirling shapes to understand whether they aligned with physical theories that describe the transfer of energy from large- to small-scale eddies as they collide and interact with one another.

I would love to have seen their grant proposal. "Yeah, we're going to study... art."

The atmospheric motion of the painted sky cannot be directly measured, so Huang and his colleagues precisely measured the brushstrokes and compared the size of the brushstrokes to the mathematical scales expected from turbulence theories. To gauge physical movement, they used the relative brightness or luminance of the varying paint colors.

One wonders if they already had a conclusion in mind when they picked those criteria, which would make it questionable science.

Huang and the team also found that the paint, at the smallest scale, mixes around with some background swirls and whirls in a fashion predicted by turbulence theory, following a statistical pattern known as Batchelor’s scaling. Batchelor’s scaling mathematically represents how small particles, such as drifting algae in the ocean or pieces of dust in the wind, are passively mixed around by turbulent flow.

And here's where most of the red flags appear, to me. Paint is, and you might want to sit down for this one, a fluid. Granted, that's just about the limit of my knowledge of artists' paint, but I have a high degree of confidence in my assertion, having seen paint in its fluid form. I've even seen paint mixed, and noted the swirls and eddies of turbulence. This is kind of like seeing milk added to coffee, and I don't drink coffee either.

Where I become more speculative is in thinking: well, he painted the thing with wet paint, so of course there's turbulence at the small-scale boundaries of brushstrokes.

Beattie agreed: “It’s an amazing coincidence that Van Gogh’s beautiful painting shares many of the same statistics as turbulence,” he said.

While there is, indeed, such a thing as coincidence, I don't agree that this is an example of it.

The study team performed the same analysis and detected the same phenomenon in two other images, one a painting, “Chain Pier, Brighton,” created by British artist John Constable in 1826-7, and the other a photograph of Jupiter’s Great Red Spot, taken by NASA’s Voyager 1 spacecraft on March 5, 1979.

Now, that seems a little more like what I'd expect from science. First, another painting; perhaps as a control of sorts. Hell if I know; I don't know that painting, and CNN neglected to illustrate it for us (there is, however, a link). As for Jupiter, we're pretty sure it exhibits all the hallmarks of turbulence on its visible surface, so it's a check on their modeling assumptions.

And yet, it shouldn't boggle anyone's mind that an artist noticed turbulence and tried to recreate it, or that one as brilliant as van Gogh was able to do it.

4 Like • Comment • View 2 comments

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