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.

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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.

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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.

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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.

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July 2, 2023 at 12:03am

#1051994

Revisited: "Epiphany"

As it's now Sunday, I'm going back in time. The destination is late 2021, and it's a response to a 30 Day Blogging Challenge prompt: "Epiphany"

I have, before I save this one, 2380 entries in this blog (the max is 3000, so you're stuck with me for a while yet). While I try not to reuse titles, sometimes it happens, especially when I can't be arsed to search for a title before I use it.

"Epiphany" was one of them. According to my search, which I did this time, I have three entries with that title (one of them includes a question mark). This annoys me, but I doubt anyone else cares.

The prompt in question, in this version of Epiphany, was to "write about a moment in your life that changed the way you view the world."

I've had a few moments like that, and it seems I wrote about most of them. The one in particular noted in the linked entry was that whether something is created or destroyed depends on your point of view about it.

That insight was an example of itself. My old worldview was destroyed, a new one created. Which one's superior? Obviously, I think the later one is. Others might have a different perspective.

Nothing since that epiphany (or since the entry where I wrote about it) has changed my mind about its truth, but, admittedly, it's not something that much affects my life. I continue to consider things like erecting a building to be "creation," while things like bombing that building to be "destruction." Likely, most people do that; it's a rare situation when you have to think about it to decide whether something is created or destroyed.

The word itself apparently comes from Greek and Latin, with the original meaning of "reveal." It's now inextricably tied to religion, as Epiphany is a celebration associated with Christmas. (It's either about the coming of the Magi, or the baptism of Jesus, depending on whose church you're talking about.) There's a lesser-known definition that's more broadly religious, meaning some sort of manifestation of the divine, not limited to Christianity.

As I'm not religious, though, that's almost never the connotation I mean when I use the word; it's the more secular meaning of a sudden insight, which anyone can have, whether they're religious or not.

As I noted in that 2021 entry, nothing is actually created or destroyed, only transformed. As others have noted, the only certainty in the universe is transformation.

Well. That, and death and taxes.

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