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:

Write_Mikey_Write!

about 22 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 24 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

1 Like

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').

1 Like

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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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