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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Brandiwyn🎶

a day ago
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.

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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').

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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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 21, 2021 at 12:03am

#1022046

Facing the Music

Music certainly has the capacity to provoke emotion in me. It's one reason I started this blog, and one interpretation of the title.

PROMPT November 21^st

Tonight/today, listen to this beautiful song.

Where does your mind go when listening to this? What emotions surface as you experience the music? If you've seen the movie (Return To Me), tell us how this affected you. Has it changed you at all?

This one, however, doesn't do much for me. Oh, I don't dislike it, not at all. It's technically brilliant, and the particular version linked above is especially well done.

Where does your mind go when listening to this?

There was a particular musical style, popular mainly pre-Beatles, that featured the "crooner." Frank Sinatra was probably the most famous of these, but also Dean Martin, who originally sang the song in question. Maybe Sammy Davis Jr. Almost all of them ended up in Vegas.

A few years ago, I got to see one of them, Wayne Newton, perform there. Amazing vocal talent. You'd never know it from listening to him speak; his ordinary voice is kind of raspy and undisciplined, but as soon as he starts to sing, it's like magic.

Anyway, Tony Bennett was certainly in that illustrious company. People nowadays dismiss it as "lounge" singing, but it's a legitimate art form. So much of music these days emanates from talentless hacks whose only draw is their looks, or maybe their dance moves (not that dance doesn't require talent, but I'm talking about the music itself). Hell, even their songs are autotuned so they don't have to work on their voice, only their display. Me, I prefer to hear songs by people who can actually sing, even if they flub a note or two from time to time; that's part of the charm.

Don't get me wrong. I'm not ragging on "damn kids these days and their music." Some contemporary music is remarkable. There was plenty of talentless tripe when I was growing up, too (but without autotune); it just tends to be forgotten, much as many of today's acts will be forgotten.

Anyway, the point is, I believe that, as with movies, it's very helpful to have a sense of historical context when considering music, be it jazz from the WW2 era, folk from the early 20th century, disco from the 70s... or lounge singers from the 50s.

What emotions surface as you experience the music?

None, really. I have an appreciation for the talent involved, but the songwriting doesn't move me. I especially appreciate the spirit of international cooperation displayed; music has, as I've noted before, the capacity to bring people together.

Mostly, because it's a couple of old guys who are excellent singers, it made me miss Leonard Cohen. Now that's music that affects me emotionally.

If you've seen the movie (Return To Me), tell us how this affected you.

Not only have I not seen that movie, I didn't even know that the movie existed until I saw this prompt.

So of course I read the Cliff's Notes versionWikipedia page. There's no way to get the emotional impact of a movie from reading the plot synopsis (which I didn't mind doing, despite spoilers, because it's in a genre I tend to avoid anyway), but I was surprised to find some really big names involved. Surprised, because as you all know, I'm a movie fan, though my participation has waxed and waned over the years. I've always especially liked Minnie Driver as an actor, and of course David Duchovny is David Duchovny.

No desire to see it, though. The plot seems trite and way too mushy, like many romantic comedies. To each their own, of course.

Has it changed you at all?

I guess you could say yes, because I now know that something (the movie) exists that I didn't know existed before a couple of hours ago. I suppose it's possible I did know the movie, but forgot about it.

Still, I hope no one interprets this entry as "that song sucks and the movie looks like it sucks too." That's not my intention. It's kind of like with beer. I like certain styles of beer. Other people like certain other styles of beer. But that diversity is awesome, because it keeps beer companies in business. Likewise, different genres of movies and music keep people making movies and music; I'm always going to prefer some to others, while other people will have different preferences.

What a boring world this would be if we all liked the same stuff.

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