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🎶

about 22 hours 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.

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').

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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.

3 Like

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

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.

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