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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Elycia Lee ☮

about 9 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

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Wordsmitty ✍️

about 22 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.

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Annette in France

a day 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.

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Elisa: Middle Aged Stik

a day 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.

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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.

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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.

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Brandiwyn🎶

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

Love it when stars align!

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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.

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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.

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Annette in France

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

I'm already out of Paris.

3 Like

January 5, 2020 at 12:04am

#972682

Ramble On

PROMPT January 5th

What music makes you want to dance?

None.

I don't use music for dancing. By which I mean, I don't use anything for dancing - my name notwithstanding, I don't dance - and music is for listening.

It took me a while to figure out that this makes me different from actual humans. I can be slow like that sometimes. When I discovered that pretty much everyone except me picks music based on some nebulously-defined quality of "danceability," I was shocked and dismayed.

I did understand that there were these events called "dances," and that only certain music was allowed at them - music, it seems to me, that has a very prominent rhythm. But to deliberately avoid meaningful music in favor of a prominent backbeat? Horror.

A musician once told me that lots of people think that's the purpose of drums in many types of popular music, but that this impression is incorrect; it's the bass line that keeps the beat, while the main purpose of drums is to back up the guitar. Admittedly, I was one of those people who made that assumption, because I was never a professional musician nor played in a band. I learned some piano when I was younger, and a piano can be a self-supporting instrument. I also learned violin, and part of that training gave me the ability to internalize the rhythm of music.

I still don't know if she was right, even though since then I've picked up some basic drumming and guitar skills. Not a lot, mind you, but some. But it changed the way I listened to pop, rock, country, blues, etc.

A while back, I took karate classes. (Wow, Waltz, that's a major nonsequitur. Shut up, I'm getting to a point here.) My sensei also taught kickboxing, so I crossed over into that, not so I could beat people up, but as a fitness thing. As part of the kickboxing classes, he'd put together a soundtrack. Now, you're probably picturing a wiry Asian guy subjecting us to K-pop or something, but no, the sensei was as pasty white Euro-American as I appear, though he'd spent years training in Japan and Okinawa, so the music was all American popular music of varied genres from the 60s, 70s and 80s.

The point I'm getting to is that kickboxing class is, I would imagine because I haven't taken dance classes, a lot like dance class. The music was there to help us stay on rhythm when doing various exercises. After one class, a young lady approached the instructor and said something like, "I don't understand this music. There's no rhythm to it. Can we have more danceable music?"

This shocked me. I hadn't had a bit of a problem picking out and sticking to the beat. But upon reflection, I realized that Led Zeppelin's "Ramble On" wasn't usually played at discos.

Which is one reason I avoided discos.

Now, if he'd played Pink Floyd, maybe that would have been another story.

So that, I think, was my moment of revelation. I'd already figured out that I was weird in that I listened to popular music more for the lyrics than anything else, but I'd never considered it to have no rhythm. To me, it did indeed have a rhythm, although sometimes a subtle one that doesn't translate well to dark rooms with mirror balls and light shows.

So, I guess I might have been hasty, there in the beginning of this entry. I do dance. I dance inside, as I do whenever music is playing. Even Led Zeppelin. Even Pink Floyd - well, except for the unlistenable experimental drug-fueled albums, maybe.

I saw a piece of sheet music lying on the piano of another professional musician once - I seem to know a disproportionate number of them, even though I can't carry a tune in a grocery bag - and looked at it out of curiosity. Another thing that makes me weird: I can read sheet music, even though I can't play an instrument with any great facility. This shouldn't surprise me, I suppose; some of my favorite musicians are just the opposite, able to play naturally, maybe not even able to read sheet music. Springsteen is one of those.

Anyway, I'm looking at this sheet music and the first thing I see is the time signature: 9/8.

Something inside of me broke.

For the uninitiated, the upper numeral is the number of beats per measure, while the lower one is the kind of note that makes up a "beat" in the notation. (For the initiated, please forgive me if I got this explanation wrong; it's been a while and like I said, I was never very good at any of this.) A more standard rhythm would be 4/4 - four beats per measure, with the beat consisting of a quarter note, or maybe 3/4, which I understand is often used for waltzes. 9/8, on first glance, just seemed to me like someone had opened a portal to the Elder Gods and let their eldritch horror into our lands.

But then I thought about it. And I realized. 9/8 is basically 3/4 time, with the basic unit being a triplet. Three sets of triplets per measure.

I can dance to that.

Leaves are falling all around
It's time I was on my way
Thanks to you I'm much obliged
For such a pleasant stay
But now it's time for me to go
The autumn moon lights my way
For now I smell the rain
And with it pain
And it's headed my way

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