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🎶

a day 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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
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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.

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September 23, 2023 at 10:13am

#1056105

Crepuscular Musings

Rather appropriate for the first day of fall...

Why Are There Three Twilights?
And we’re not talking about vampire books.

Yes, I said fall. I get really, really tired of saying "or autumn if you're in an Anglophone country that's not the US, or spring if you're in the southern hemisphere, but some cultures considered this to be the midpoint of the season, so it gets complicated." As mine is the only perspective that matters, it's fall, and today sucks because we're getting spitting rain thrown off from a minor hurricane.

Anyway. Why is it appropriate? Because while "equinox" literally means "equal night," implying that day and night are of equal length, that's not technically the case on the equinox, and part of the reason is twilight.

For example, this is the (badly formatted) almanac for my area today (courtesy of Weather Underground):

Sunrise 7:03 AM Sunset 7:11 PM
Civil Twilight 6:37 AM 7:38 PM
Nautical Twilight 6:06 AM 8:08 PM
Astronomical Twilight 5:35 AM 8:40 PM
Length of Visible Light 13 h 0 m
Length of Day 12 h 8 m

"Length of visible light" takes twilight into account. "Length of day" is, you'll note, 8 minutes longer than it would be if day were actually equal to night. That's because the atmosphere refracts sunlight, so the sun appears to rise earlier and set later than it would if we didn't have an atmosphere, which, if that were the case, there would be no one around to be picky about these things.

The three twilights there... well, that gets us back to the Atlas Obscura article.

When the sun slides out of sight—or, more accurately, when Earth rotates so that your particular place on the planet is no longer exposed to our day star—but before darkness consumes the landscape, there is the magical, muted time of twilight.

One of my all-time favorite words is "crepuscular." Animal activity can be crepuscular, or there might be crepuscular rays visible in the sky due to distant clouds. It's an adjective meaning "of or pertaining to twilight." I'm unaware if it has a noun form; "crepuscule" just sounds wrong.

Most of us think of twilight as a single period of transition from bright day to deep night, but there are actually three twilights: civil, nautical, and astronomical. Each is defined by how far below the horizon the sun is, and hint at our lives before artificial light and GPS.

Or, for those of us who care, our lives now.

Civil twilight has nothing to do with being polite; it’s simply the evening’s first twilight, which starts at sunset: the moment when the sun’s center is exactly at the horizon or, to get technical, 0 degrees below it.

Again, this is not precisely true because of that refraction thing. It also happens, in reverse, near sunrise.

Once the sun is 6 degrees below the horizon, nautical twilight arrives. If you’re not a degrees kind of person, hold your arm out to the horizon, with your hand turned so your palm is facing the horizon and your fingers, parallel to the horizon, are together but extended. Close one eye. The width of your first three fingers is roughly 5 to 6 degrees, depending on the size of your hand.

That's nice to know and all, but at that time, you can't bloody see the bloody sun, so how do you know how far below the horizon it would be without math and a clock?

Nautical twilight is the period after sunset when, though no longer able to read a chart without a lantern or torch, sailors could still take accurate readings.

Reverse that for morning. Also, this doesn't take into account cloud cover or moon phase, which can shorten or lengthen the time when one could see the horizon. Also also, things are a bit different on land because you sometimes have hills and mountains and stuff. Your own height of observation plays into it as well.

Regardless of all these variables, it's still cool stuff to know.

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