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.

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

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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

January 13, 2021 at 12:02am

#1002003

These Are The Good Old Days

PROMPT January 13th

What songs hit you with a wave of nostalgia every time you hear them?

I... don't really do nostalgia.

Not on a regular basis, anyway, and not tied to songs. Which is not to say that I don't sometimes miss some aspects of the past; it's just that I use music differently, I suppose.

Sure, there are songs from my younger days that can trigger memories, both pleasant and not so pleasant. But I also enjoy some newer music, and some older music that I somehow missed along the way. Music, to me, is music: I don't care much about what "decade" it was made in (except insofar as my mental metadata about a song usually includes what year it was released), and while songs have a genre, that's often secondary to me. As long as it's a good song -- subjectively and/or objectively -- I generally want to hear it again.

In other words, unlike I suspect most people, I'm not trapped in any one period of popular music, or any one type. I've been known to create mix tapes, back when mix tapes were a thing, with folk songs back to back with heavy metal. Or punk followed by glam rock. From vastly different time periods. For example.

Now, obviously, I'm often drawn to familiar music like just about anyone, but, again, that's not usually because of any associations it has, but because I simply consider it quality music.

I hope I'm never one of those "all music these days sucks" types. Sure, there's a lot of music out there that is utter crap, but I distinctly recall that this has always been the case. It's just that no one ever plays the crap music from when I was a kid anymore... because it's crap. They still play the new music that's crap because it's new, so I understand why people might draw that conclusion. (It's the same way with books.) Admittedly, I haven't really been keeping up with new music, lately, but every once in a while I'll hear one at the taphouse or something and, curious, I use the handy "Hey Google, what song is playing?" feature on my phone, and often, it'll be something from the last decade... or maybe from the sixties.

And can I just say, that shit is sorcery -- the music recognition search thing. It picks up like three measures of the music and comes back with song, artist, album, year, label, producer, lyrics, and pretty much anything else I'd want to know.

Perhaps that's why I don't often wax nostalgic: apart from certain political and epidemiological problems going on in the US right now, I feel like I'm living in "the good old days." Like I said, yeah, there are things from the past that I miss: my parents, old friends, cats, being able to travel, that sort of thing. But damn if technology isn't pretty fucking awesome right now. Like, if I want to hear a song? Comes right up on my phone and I Bluetooth it to a high-quality speaker.

Sorcery.

8 Like • Comment • View 8 comments

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