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.

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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.

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.

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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.

2 Like

March 17, 2023 at 2:19pm

#1046645

Wrong

Heading back to "JAFBG"

[XGC] for a random prompt...

Tell us about a time when you were proven wrong.

Me? Wrong? Unpossible.

Okay, no, that's a joke. I'm wrong a lot. Sometimes even right here in this blog. I'm an adult, though; if you can show me I'm wrong, I'll swallow some corvid and admit it.

There's lots of things I could be wrong on, but until shown otherwise, I'll stick to my opinions. Things like the absence of free will, the arbitrariness of the Gregorian calendar, the reality of time and space, the mistaken definition of a Blue Moon (which is related to that Gregorian calendar thing), the absence of space aliens on or around Earth, the dangers of evolutionary psychology, and much more—all of those things are topics that I've discussed in here, that I have a high level of confidence in. But, say, if a flying saucer landed in my street and disgorged a three-tentacled, seven-eyed mauve creature with a tricorder and universal translator, well, I'd change my mind about aliens real quick. Assuming that I haven't decided to experiment with magic mushrooms, that is.

Worse, though, are the things I'm absolutely certain of. Those are dangerous, because, like anyone else, I get stubborn, and invested in my ignorance.

Let me just come up with an example off the skin of my ass:

Long ago, when I was married, my wife bought a shower curtain. This is not such an unusual thing, I think; I'd been using the same shower curtain for way too long, and I can own that. Fine. This particular one didn't even have any designs; it was just the clear plastic kind that hangs down inside the tub to keep most of the shower water inside.

She also bought a bottle of, I don't recall the brand name, Tilex or something. Used for spraying down the shower tiles after use, supposed to keep mold and mineral scale from showing up. Again, something I'd been neglecting. What can I say? I lived (basically) alone before she moved in. Had housemates, but also had another bathroom for them.

So I proceeded to, every time I took a shower (which, contrary to what you might have inferred from all of the above, was every day), spray down not only the tiles but also the shower curtain, on the (what I thought at the time was a) perfectly sound principle that it would keep us from spending money on another shower curtain for a while.

She found out that I was doing this and we got into an argument. "We can just buy a new shower curtain" was basically her position on it. I disagreed; a shower curtain, like any piece of furniture, should last a lifetime. That was my position.

It was years—years—later that I had an epiphany:

A bottle of Tilex (or whatever) was like five bucks.

A clear plastic shower curtain was like ten bucks, and takes about five minutes to install.

The epiphany was that all the Tilex I had been blasting on the curtain would end up costing more than it would to, as she suggested, simply replace the shower curtain every year or so. So, from a purely financial perspective, I was completely, utterly, incontrovertibly, wrong. Now, I could still argue that, environmentally, I was in the right, what with all the plastic waste and all, but to be perfectly honest, that didn't enter into my calculations.

Thing is, by then, we'd already separated, and I didn't feel the need to confess my wrongness to her at that point. These days, I use two shower curtains: the inside clear plastic one which gets replaced every year or so (environment notwithstanding), and an opaque cloth outer one. I'm not even sure why; I don't share the bathroom with anyone, so it's unlikely that somebody would come in while I'm showering. I just like the design, which has cats.

That epiphany, though, made me wonder: what else am I sure of that just isn't the case? As I'm no longer married or otherwise partnered up, there's no one to set me straight on these things. Not even my cats. The only time they complain is when I'm five minutes late with their dinner.

So, yes, I was wrong. And probably still am, but if so, I wouldn't know it.

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