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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 15 hours ago
In response to "Extro Special"

I'll also use the spelling extrovert because if I use extravert my spell checker flags it. In college psychology our professor termed his own words, innervert and outervert because he thought they fit better. I, however, thought they sounded like definitions of belly buttons.

1 Like

Kåre เลียม Enga

about 21 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

Dec 26, 2024 at 4:20pm
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
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?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

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

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
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

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

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

1 Like

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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