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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10 Most Recent Blog Comments:

Writer_Mike

about 6 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 22 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

a day ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

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.

6 Like • Comment • View 3 comments

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