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:

Wordsmitty ✍️

about 11 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 14 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 16 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

Lilli Munster 🧿 ☕ 🎃

Sep 24, 2024 at 9:10am
In response to "All Out of Louvre"

Have a wonderful time in France. Have you run into Annette in France yet? She's there too!

2 Like

June 9, 2020 at 12:21am

#985316

Blame it on the Rain

This will be entry #3 for "Journalistic Intentions"

[18+]

9. You have to look through the rain to see the oncoming bus, dump truck, and motorcycle.
Solace.Bring

I'd probably have gone with "clouds." You have to look through the rain to see the clouds.

I don't really know the original quote here, but I suspect I would be disdainful of it. As it is...

Well. I go through optimistic phases, but I'm not what you'd call an optimistic person. There's a scene I vaguely remember from Doctor Who, in the Matt Smith incarnation. I don't really remember the situation or the exact wording, and I can't seem to find it anywhere because most people associate the Doctor with optimism, but in this case he says something like, "It's not like things could get any worse." Things proceed to get worse, and then he says, "There are some sentences I should just stay away from."

In other words, it's always darkest just before things fade entirely to black.

I mean, obviously, I know it's not true. Simple statistics show that if something unpleasant happens, there's a good chance that something nice will happen afterwards, in addition to the possibility that things will get worse or that they'll stay about the same.

But, see, that's not helpful. I like surprises, but only pleasant ones. If I find an unexpected creature in my home, it would be a nice surprise if it were a unicorn, and a lousy surprise if it's a rabid coyote. So if I expect the rabid coyote and then I get the rabid coyote, I can be like, "Yep, I was right," and I'd feel good about being right all the way up until the time I'd have to get rabies shots. But if I expect the coyote and get the unicorn, well, that's a pleasant surprise. On the other hand, if I expect the unicorn and get the coyote, that's all kinds of disappointing unpleasantness.

So, yes, I'm going to expect the worst, because that way, I can feel good about being right, or I can feel good about the pleasant surprise. One thing I can never be is disappointed.

Well, that's the theory, anyway. In practice, I get disappointed quite a bit, so I just keep becoming more negative until I'm no longer unpleasantly surprised.

So to get back to the remixed quote above, I find rain to be generally pretty neutral. I know people who love it or hate it, or maybe it's situational, but to me it's like, well, I don't live in a desert (which makes me wonder just why all of these rabid coyotes are running around) and so I have to expect it to rain sometimes. It does no good to love it or hate it; it just is.

But it does tend to put a crimp in one's plans, and when that happens, I can only expect things to get worse, like I get caught in a three-way weather-induced accident between a bus, a dump truck, and a motorcycle.

And when that doesn't happen, I feel pretty good. When it does happen, I'll be dead.

Yesterday's entry included a mini-contest. The challenge was to describe the smell of coffee, because some stuffy German philosopher once said it couldn't be done.

And there were some great responses. All of them, actually. You guys can't make this easy on me. But as per the above, I expected that.

Based on the responses, I'm starting to think Wittgenstein might have been onto something after all. You can compare the aroma of coffee to some other scent, which is useful but also kind of kicks the can down the road; how would you explain these other scents? You can use metaphor, as in "the liquid essence of a black sun," which I thought was particularly poetic, but none of us have experience with black suns. And then there's the neurological route, as in "triggers the pleasure centre" (which I disagree with, but as I said, I'm not going to dismiss an attempt just because I don't like the aroma myself), but while that's accurate (for people who aren't me), it doesn't really convey the sensation.

But maybe there's really no way to convey it on a practical level. I used the example yesterday of describing "blue" to a person blind since birth, but I suspect that trying to describe the scent of coffee to someone who has never had a sense of smell would be a problem just as intractable.

As I said, I thought everyone provided a great response, but I'm going to go with the one that made me laugh.

How would a non-coffee drinker describe the aroma of coffee? Well, like this. Coffee's aroma is the closest we'll get to someone trying to set moist soil or peat on fire using a non-petroleum based ignition fluid. You could imagine something like vodka, 151 rum, or a very small amount of C-4 in this scenario. These ignition fluids can account for any trace scents that may appear against the soil-based backdrop. If it's sweet, blame the rum. If it's clean, blame the vodka. If there's a flinty or mineral note, well, C-4 could account at least for part of that.
Elisa: Middle Aged Stik

, I owe you a Merit Badge.

This is fun. I'll keep doing it from time to time as long as I keep getting enough responses to make it worthwhile, so everyone will have another chance soon.

6 Like • Comment • View 6 comments

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