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

October 8, 2023 at 9:05am

#1056983

Revisited: "Another Avocado Article"

Today's random rerun comes from just over a year ago, which pushes very close to my self-imposed one-year exclusion: "Another Avocado Article"

Because it's a year old, and based on an article a year older than that (the link is still valid), there's not much to comment on. But I did find a few things, all quotes from me.

Does the pumpkin seed paste come with pumpkin spice? I'm asking because, though the article is from last November, it's almost the autumn equinox here and now—peak pumpkin spice season. And I'm not immune: I just picked up a six-pack of Atomic Pumpkin Voodoo Ranger, a seasonal beer offering from New Belgium out of Colorado.

You know, I completely forgot to pick up APVR this year. It simply didn't occur to me. I did buy a couple of packs of Pumking, the Imperial pumpkin ale produced by Southern Tier out of New York. Excellent beer.

They also make Warlock, which is a pumpkin stout; and a caramel Pumking, which is not nearly as good as it sounds.

Anyway, maybe the Atomic Pumpkin Voodoo Ranger is still available. I'll have to check.

Eh, whatever. I rarely eat guac anyway.

I didn't get into the reasons for this, but it's not that I dislike the stuff. Depending on how it's made, it can range from okay to delicious. It's just that I'm too lazy to make it myself, and restaurants tend to price it like an appetizer.

That's how you get people to stop consuming something: price it out of their range.

Now, just to be clear, I'm not arguing for or against avocados. Personally, I can take them or leave them; the avocado industry could dry up (pun intended) tomorrow and I doubt I'd miss it much.

I don't think this came off as I intended. I like avocados (but only the Hass variety), and I use them occasionally; I just don't make guacamole. I'm pretty sure I would miss them if they disappeared, if only because the surest way to ensure I want something is to make it so I can't have it. Still, they're not exactly what I'd call "essential," not like beer, pizza, or ground beef. (Avocados can be really good as a hamburger topping, but they suck on pizza.)

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