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

November 19, 2021 at 12:04am

#1021917

Epiphany

I'm pretty sure I've written about this before, but it's been a while and I've been thinking about it a bit recently.

PROMPT November 19^th

We've all had one or several epiphany's in our lives. Tonight, write about a moment in your life that changed the way you view the world.

And I touched on it in this week's Fantasy newsletter, but most people don't bother to read editorials. I know I don't.

I've heard people describe creation and destruction as opposites, or as two sides of the same coin.

But one night, driving alone (which is one reason I like to drive alone), it hit me like a truck: they're not opposites, they're not two sides of the same coin... they are the exact same thing.

Of course, there is one distinction, and that's local entropy. As we all know, the total entropy of a closed system has to stay the same or increase (you do know that, right?) But the key there is "closed system." We can construct a building, for instance, which has lower entropy (at least at first) because we're putting energy into it, causing entropy to increase elsewhere.

So for what I'm talking about here, entropy is local and irrelevant. I'm talking about the inability to make a truly objective decision as to whether a thing or system was created or destroyed. Any such label applies only to the value we put on the things that have changed.

Take the building I just used as an example. You can speak of creating it, building it, which is usually a massive coordinated effort between many different specialists, not just onsite, but all of the people who make the individual items that go into the building's construction.

So you chop down a tree, which destroys the tree -- but you make lumber from it. One could even say that putting the lumber together into the building's frame destroys it; it's no longer lumber but studs or joists or whatever. The value we put on what is created or destroyed is subjective.

One might object, "sure, but if you drop a bomb on something, it's destroyed." Yes, it destroys the bomb and the target, from one point of view. From another, it creates debris and rubble and... well, at the risk of getting morbid, maybe corpses.

Sure, the people getting bombed would consider that a bad thing, but to the ones doing the bombing, that's the desired outcome. They've created what they set out to create, as chaotic as that is.

Right now, I'm creating a blog entry, but in order to do so I'm destroying the previous arrangement of atoms on a semiconductor chip. You can't write a book without destroying blank paper.

Now, before someone comments about it, yes, I'm fully aware that in the larger picture, nothing can actually be created or destroyed, only transformed, but that's not the point, either. The point is, like I said, "creation" and "destruction" are mere value judgements slapped onto the larger concept of change.

And that was the epiphany I had maybe a quarter-century ago for no particular reason except that I was driving, it was night, and it was my birthday.

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