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.

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.

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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.

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.

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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.

2 Like

April 29, 2023 at 8:26pm

#1049011

Stuck in Committee

I found more new-to-me breweries today.

But let me start by recapping yesterday.

The town I was in was, if you care to look at it on a map, Wytheville, VA. I'd never spent any time in it before, but it's exactly what I expected from a small town in the mountains: two stoplights, 500 people, 600 churches, and a rustic nonfunctional clock on the courthouse.

And, as I noted, two breweries. Here's the funny part, though: as far as I know, the breweries aren't related. But one is called 7 Dogs, named after seven dogs the owner had rescued (with, fortunately, a nod to the rescued cat who kept the dogs in line).

The other is named Seven Sisters. Yes, this is important because they both had the number seven in the name. But as I approached this second brewery, I pondered: would it be a reference to actual sisters, or to the star cluster variously known as the Seven Sisters, the Pleiades, and Subaru? Because as you know, I'm a sucker for anything astronomy. And I drive a Subaru.

So I was pleased to see a nice big print of the Pleiades as I entered.

And yet, the beers were named traditionally women's names. Like Rosie, Julia, Edith, etc. Which is fine; it's good to have a theme. But one thing disappointed me: there was no beer named Kate.

You see, I like to order tasting flights, which usually consist of 4-6 sampler-sized beers. The missed opportunity was that I could have had my Kate, and Edith too.

I will just pause while you absorb the greatness of that joke.

Ready?

Okay. So, after passing out and sobering up, I spent the better part of the day driving across Virginia, to the east, all the way to the Richmond area, as prompted by my random number generator. Nice drive, great weather. I don't get to the southern part of my state very often, and never before along an east-west route. It could just as easily have landed me back in West Virginia, or Ohio, Kentucky, or Tennessee. Or maybe even a Carolina. But instead, it kept me in Virginia. Fine. Such is the nature of randomness.

Here, I found a couple of breweries in Chester, which is a city south of Richmond but north of Petersburg. No, it confuses me, too; as far as I'm concerned, all those cities and everything else in the vicinity is Richmond. There really ought to be a rule: unless there's a significant river (the James doesn't count) or a lot of trees and/or farmland in between, just fucking merge the cities.

If you still don't know what the hell I'm talking about, try Google Maps again. I've been drinking.

Just one thing of note on the drive:

I saw a wake of buzzards.

By "buzzards," I mean turkey vultures, cathartes aura, my namesake and the closest thing I have to a spirit animal. It's not unusual to see turkey vultures around here, but usually, they're loners. This is one reason why they're my spirit animal. Sometimes, you'll see a bunch ("wake" is apparently the official collective noun for turkey vultures) of them hanging around, but that's rare. So it surprised me when I saw not two, not three, but five of them just chilling together. I had the windows down, but I didn't see or smell any carrion in the vicinity, so maybe they were just having a chat. Nonsocial birds being social.

Now I see that they're only called a "wake" when they're munching on a carcass. As these weren't, there are various possible collective nouns.

"Committee" seems to be the leading candidate. A committee of turkey vultures.

But I digress.

Did the Universe send me a committee of vultures as a Sign? A Message? Well, no, of course not. But that doesn't mean I can't remark on how cool and unusual it was to see a committee meeting.

Ugly-ass birds, but they're magnificent anyway.

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