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
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.

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January 25, 2021 at 12:02am

#1002747

Navigation

PROMPT January 25th

Write about something antique or inherited that you own. Who owned it before you? Where did it come from? What’s its story?

Unfortunately, I have quite a few inherited items - unfortunately, that is, because my parents have died.

I've talked about some of them in here before, I know, but I can't really remember most of them. I do remember writing about the barometer.

But I can't recall if I've discussed my dad's sextant

or not. Oh well, what the hell, I just fixed myself a martini, so sextant it is.

These days, of course, sailors have other means of navigation, mostly GPS. I have a vague idea of how that works, having used it myself and looked into the (very interesting) technology behind it. What's most interesting about GPS is that if you don't take general relativity into account, it loses precision remarkably quickly. It absolutely relies on science that people in the 19th century couldn't even have imagined, let alone understood. Well, to be fair, if you took the time to explain it to many of them, they'd get it; we haven't gotten any smarter; we've just increased our understanding and changed our technologies.

Still, for the greater part of the 20th century, they understood the principles, but it wasn't until around the turn of the 21st century that GPS became widely available. So as far as I know, a sextant is something that's only about 20 years behind the times. I could be wrong about this. Martini, remember? And so I can't be arsed to look anything up. Just don't take anything I say here as the absolute truth. In vino veritas, but in gin, whatever.

So a sextant is largely obsolete. I like to think that serious sailors keep one around for emergencies, but from what I understand, it's not very useful without two other items: a chronometer and an ephemerides. And in any case, I'm not a sailor like my dad was, so I don't have any actual use for it.

A chronometer is mostly just a fancy word for clock. When mechanical clocks were invented, they relied on a pendulum, a thing that provided a predictable periodic "tick." These were completely useless at sea, what with all the waves and shit. So the big problem in intercontinental navigation was to invent a chronometer that relied on something other than gravity -- but I'm getting ahead of myself. Let me back up.

But first, I'll tell you what an ephemerides is. It's a table of where a certain heavenly body is expected to be at a certain time. These calculations are fairly complicated, but at the same time straightforward. You could have one for the sun, the moon, Jupiter, or any of the other planets or stars.

Now, backing up.

Navigation requires at least four pieces of data. 1. Latitude. 2. Longitude. 3. Heading. 4. Speed. There are probably others, but... gin. Oh yeah. 5. A freakin' map.

Latitude, at night at sea in the northern hemisphere, is dead easy: 1. Find Polaris, the North Star. 2. Determine the angle between Polaris and the horizon. 3. That angle is your latitude. (Step 2 requires an instrument such as the sextant.)

Finding longitude, on the other hand, is complicated as fuck. You have to know the time, and you have to know the expected location, in the sky, of some star or planet or some such. Knowing the time is where the chronometer comes in; knowing the expected location of a certain point of light is the job of the ephemerides.

Heading and speed are largely irrelevant to this description, so I'm going to drink more gin and ignore them for now.

There.

So. You know your latitude because it's night and you've shot Polaris with the sextant. And now you know the longitude, because you know what time it is (or, rather, what time it is back in London or whereverthehell) and you have star charts so that you can tell the difference between where, say, Sirius would appear in the sky from London and where Sirius looks like to you on the heaving deck of a ship.

The reason you know these things is because you have the sextant to determine the angle between the star (or whatever) and the horizon.

There's also a way to "shoot the sun;" that is, figure this shit out in the daytime. This is above my pay grade, even if I weren't three sheets to the wind right now (that's also a nautical phrase, by the way, in case it wasn't completely obvious).

People talk about a "moral compass," what they use to determine their direction in life. A compass is another important tool in navigation (see: "heading"), but it's not the only important tool. I keep the sextant around for two reasons: because it's a constant reminder of what my father lived for a good part of his life (not most; he was a sailor for about 1/4 of his 90 years), and also because it's a reminder that you always need to know your location. Metaphorically speaking.

I told you the other day that I've been going through all the episodes of Star Trek. Picard kept a sextant in his Ready Room -- and given the utter uselessness of a sextant in interstellar space, I like to think it was for the same reason I keep one: that you should always know where you are.

And where you're going.

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