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:

Elycia Lee ☮

about 5 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!

2 Like

Wordsmitty ✍️

about 18 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 21 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 23 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

March 5, 2020 at 12:02am

#977166

Moonlighting

Today's entry is for:

Journalistic Intentions

(18+)
This is for the journal keeping types that come to PLAY! New round starts July 1!
#2213121 by Elisa: Middle Aged Stik

What is a "reactive visualizer?"

That was my first question, as I haven't encountered these videos before. The YouTube description is mute (see what I did there) on the subject, and I long ago trained myself to never look at YouTube comments.

And even after Googling, I'm still not sure. It looks like a "Piano Hero" attachment that drops the notes that you're supposed to strike?

But that was after watching the video. During, I was too mesmerized to care much about the technology.

Speaking of technology...

It turns out I have a fondness for things that take things from the past and give them a new twist.

It could be argued that just putting Beethoven on the internet does that, or giving out mp3s of singing monks, or whatever. But the medium itself is irrelevant; if it's just some dude sitting at a piano and playing a sonata, well, that's been the same for a few centuries now.

I once saw a video - I'd link it, but I don't want to overdo the embeds, so you'll just have to look for it if you're interested - of a rendition of (speaking of ol' Ludwig) Ode to Joy. Now, of all of the overdone Beethoven pieces, that's the overdonest, but it's arguably the second most recognizable tune from that composer (the first being, of course, the triumphant FUCK YOU of the opening notes of the Fifth). But this one had a twist. No, a few twists. First of all, it was an all-Japanese orchestra. If memory serves, it was also all-female, but don't make me swear to that because I can't be arsed to find it again now. Second, it was an all-theremin

orchestra. And finally, they played a version that was in Ragtime style.

Bottom line: a Japanese orchestra playing the music of an Austrian composer using 20th-century Russian technology, in an American style.

Or - and this one's my favorite - a mashup of electric guitar covers of one of the earliest known pieces of sheet music, Pachelbel's Canon in D.

To stray from music and into theater, I recently saw a video of a stage production of Much Ado About Nothing with David Tennant and Catherine Tate. That's right - The DoctorDonna, on stage, doing Shakespeare. It's glorious. Go find it.

Point being that modernization isn't a bad thing -- provided you know where the shit comes from in the first place.

As far as I'm concerned, this is why the internet is great. Well, that and cat videos. And maybe porn. No... cat videos.

Anyway, back to the video up there at the top.

I'd also never heard of Rousseau. Don't look at me like that; I appreciate classical music, but my heart belongs to rock. Some say it's made of rock, and who am I to argue? So I looked him up, briefly, but all I saw is a bunch of other reactive visualizer videos. Which is fine; I'll probably see more of them at some point, especially now that YouTube's algorithm has surely decided to inundate me with suggestions for other Rousseau videos. That's fine. It's something different from my usual suggestions of math / science explanations, rock and roll, cat antics, and My Little P- I mean, er, cooking vids.

And now I see that one of my suggestions is a video entitled "Does Quantum Immortality Save Schrödinger's Cat?" I haven't clicked on it yet, but I'm hoping it has a cookin' soundtrack.

3 Like • Comment • View 2 comments

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