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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Write_Mikey_Write!

a day 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.

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Adherennium Plotting something

a day 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"

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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.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

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Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
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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').

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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!"

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June 12, 2019 at 1:35am

#960659

Complex Numbers

A good barbecue always has some great music...make a five- to ten-song playlist of some perfect summer tunes to play for your party. And/Or, if you could pick any one band/artist to play in your backyard, who would it be?

You probably don't want me picking the barbecue music.

I've noticed that for most people, happy, upbeat music makes them smile, while sad songs make them depressed. For me, sad songs make me smile, while happy music makes me stabby. My ex-wife used to give me shit about this. Happy songs also seem to me to be very simple, while I appreciate the complexity of more thoughtful music.

I'm also very lyrics-focused. It surprised me when I found out that some people don't even listen to lyrics. To me, they're the point of the song. Songs without meaningful lyrics - no, they don't have to be deep, just something other than nonsense - just don't register for me. Having very little musical talent myself, I don't always pick up on the melodic nuances in a song. Usually, I can tell if it's in a major or minor key, and the genre of the music, and sometimes I can even tell what instruments are playing. But words? Words are kind of my thing. I spent one summer analyzing the lyrics to American Pie, for example, and I think I've even figured them out.

So for a cookout, I'd probably pick songs about the end of summer, or about being just about anywhere else. I noticed one day while on a road trip that I have a disproportionate number of songs about travel on my playlist. I didn't plan this; it's just that many of my favorite songs involve being or going somewhere.

This will probably put a damper on everyone else's good time, so maybe let someone else pick the songs. Most people seem to think it should simply be background noise, while I actually care about the tunes themselves.

Meanwhile, let me give you an example of the kind of thing I'd play, and you can decide for yourself.

As an aside, I had a... moment... today. As I've mentioned before, I've managed to go to the gym every day (save for a week when I was out of town) for several months, now. Mostly, it's a chore, but I do it because I started to feel some improvement, and it's supposed to be good for me. But that doesn't mean I like it. Far from it. It's an exercise (pun intended) in trying to do unsatisfying things that might lead to a better life for me.

Today, though... I leveled up. That is, I was able to notch up the weights on the machines. This made me feel pretty good about things. But then the gym's music system cued up Stevie Nicks' Edge of Seventeen. I don't know if it was her voice (I've always wanted a windup Stevie Nicks to sing me to sleep every night), or the relentless 16th note guitar, or the familiarity of it, but it made me feel... strange. I thought about what emotion I was feeling, running through my usual repertoire: depressed, sad, guilty, schadenfreude, irritated, frustrated, blank, anxious, weary, etc. None of them fit. And then it hit me: this was a feeling I usually only get after a couple of beers, but I haven't had anything with ethanol to drink for a week.

Happy.

I was fucking happy - if only for a few moments.

Later today, I have a doctor appointment scheduled. That should fix that.

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