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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Write_Mikey_Write!

about 10 hours 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.

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Adherennium Plotting something

about 12 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.

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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.

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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.

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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.

February 16, 2024 at 9:54am

#1064272

Violet

Here's a colorful entry for "Journalistic Intentions"

[18+]:

Violet

Try to imagine a color that doesn't exist. Go on; give it a good shot.

If you're imagining "violet" because that's what I put in the title, well, no. Insofar as color exists at all, violet is on the spectrum at around 400 nanometers.

People hardly ever describe something as "violet," though, unless they're talking about the flower, in which case, according to the ancient rhyme, violets aren't violet, but blue. Actually, I'd wager that there's far more mention of violet's neighbor, ultraviolet. Nope, that's not a color that doesn't exist; we just can't see wavelengths that short. It exists, but not to our eyes.

Ultraviolet is such a common word that we don't often stop to think about how cool a word it actually is. Ultra. Violet. Should be a superhero name. "Scatter! Ultraviolet's here!" She'd be way cooler than her archnemesis, Infrared. (That's a deliberate pun on a couple of different levels and I'm quite proud of myself for it.)

Getting back to the blueness of violets (the flower), though, I'm sure you learned the mnemonic for the official colors of the spectrum: ROYGBIV. Often—including on Pride flags and iconic album covers—poor indigo gets left out, leaving just six colors. Which is too bad, because indigo is probably just as cool a word as violet. Much cooler than red. Or yellow.

There exist, of course, not just seven colors in the spectrum, but a whole... well... spectrum of them. Red only gradually fades into orange, which only gradually fades into yellow, and so on all the way to the invisible end of violet.

Why do we say there's seven, then? Well, I'm pretty sure we can blame Isaac Newton for that.

In addition to pretty much inventing calculus, science, gravity, and motion (or at least the way we think about these things), Newton did a lot of study on the properties of light, a topic that later generations of physicists would be absolutely obsessed over. But Newton was, like everyone, a product of his time, and he was also greatly intrigued by mysticism.

So, I can only assume, when he shone sunlight through a prism to create an artificial rainbow (as reproduced in a certain classic album cover), he decided that there had to be seven colors. Because there were seven planets, see? And each one ruled a different day of the seven-day week. In mysticism, the sun and moon counted as planets, because they didn't know any better. If you're wondering, it went Sun, Moon, Mars, Mercury, Jupiter, Venus, Saturn. Some of those are obvious in English; others are obvious if you know some French: mardi for Mars, mercredi for Mercury, jeudi for Jupiter (Jove), and vendredi for Venus. English adopted the Germanic interpretation for some of the weekday names, which is why we get Norse god names instead of Roman ones.

Back to Newton, though. Researching this entry led me to this site,

which is enlightening (pun absolutely intended). Where was I? Oh yeah, seven colors. One major point of what Newton did was his invention of the color wheel, where the seven principal colors he identified wrapped around into a circle, with red touching violet. Okay, that last phrase sounded way less naughty in my head. Anyway, Newton didn't have all the information we do about light's wavelengths, or even its dual wave/particle nature (though he laid the groundwork for that discovery, centuries later). Now, we know that prisms (and rainbows) work because the bending of light in refraction, through glass or through suspended water droplets, depends on the wavelength of the light. But my point is, red and violet are, despite their proximity on color wheels, at opposite ends of the visible spectrum. And there's a lot more invisible spectrum than visible: gamma rays, microwaves, radio, etc.

All those invisible (to us) wavelengths are real.

But you know what's not real, that doesn't exist anywhere on the spectrum?

That mixture of red and blue pigments that we call purple.

And that's the answer to the riddle I started with: Purple is a color that doesn't exist.

Well. A philosophical argument can be made that no color actually exists. This is related to the holes thing I did a few days ago. But on the spectrum between "definitely exists" and "definitely does not exist," purple is closer to the latter than violet is.

Violet is as close as we can get to purple and still be able to identify it in the sun's radiance.

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