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:

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.

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').

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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!"

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

1 Like

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

February 4, 2021 at 12:05am

#1003605

Immigrant Song

Entry #1 of 8 for "Journalistic Intentions"

[18+]

Immigrant Song

Ready for some music theory?

No? Well, too bad.

Now, look, I'm not entirely ignorant of music theory, but some of the shit in the video at that link is way beyond me, kind of like how I have some knowledge of physics but then they start talking about quantum this and dark matter that and my brain shuts off.

The analysis that guy goes through is the musical equivalent of literary analysis. And as with literary analysis, I think they often read into the art things that the artist didn't consciously intend. "Why did you play that particular note at that time?" One might expect the musician to launch into a detailed explanation of harmonics or dissonance or setting up a musical resolution or some shit, but chances are they just go, "Well, it sounded right."

Making it sound right is the musician's job. Explaining why it sounds right is best left to theorists.

As a writer, I've always been more focused on the lyrics than the music. As powerful as the tune is, I have to admit I dismissed it for a long time because, come on, enunciate, Plant. But of course when the internet came along, I was finally able to decipher the more obscured words. Also, apparently, "Waa-aa-aaaaaaaaa-AHH! is a word. That's an official Robert Plant lyric right there.

Probably the most awesome use of the song in its 50-year history (don't you feel old now?) was in Thor:Ragnarok, and the decision to use it not once but twice in the same movie, for whatever reason (I can no more analyze movies than I can music) just works.

This is how you choreograph a fight scene to battle music.

Few musical selections have fit the tone of a scene, or an entire movie, so well.

Of course, no discussion of Immigrant Song would be complete without the inclusion of this seminal music video:

As someone notes in the comments (yes, I ventured into that swamp): "The internet peaked when this was originally posted. It's been downhill ever since."

Mini-Contest results!

Seven responses, all relevant to the question. As I noted, I'm just picking one at random, but I appreciated all of the comments.

The Virtual Dice returned a 4, so the Merit Badge goes to... (drumroll) (Hey stop with the bass line, I said drumroll)... Graham B.

!

We'll do this again soon.

5 Like • Comment • View 2 comments

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