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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 4 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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').

1 Like

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.

August 16, 2022 at 12:03am

#1036547

Probe

Another Cracked article, this one tickling my confirmation bias, so I'm sharing it.

Why Alien Abduction Stories Are BS, Even My Own

I know I've written about this sort of thing before. Here's one from eleven months ago: "Out There"

. Short version: I note the similarities between alien abduction stories and sleep paralysis, because I've experienced the latter far more than I'd like (that is, more than 0 times).

If you’ve followed my work here on Cracked or ever had me corner you at a houseparty, you know that I have a complicated relationship with the supernatural. It’s something which I am deeply fascinated by, desperately want to be true, and am also completely convinced it’s all B.S. of a Q-Ray Bracelet order of magnitude.

I can relate, except for the "desperately want to be true" part. I want explanations, not confirmations.

Now, I'm not going to do a lot of article pasting today. I just don't have the time. But the article goes into one of the most famous alien abduction stories, Betty and Barney Hill.

Then,

Abductees generally report a sense of missing time, strange lights, looming ominous presences, and, most notably, some sort of bodily violation, usually with metallic instruments. This is the famous ‘anal probing,’ which when you think about is really bizarre. When we humans discover a new species here on earth, our first step isn’t usually to jam stuff up its butt.

Never had that happen to me, but then, I find I'm way less fascinated by butts than most humans. I was just reading yesterday about some guy who went to the ER with a hot water bottle rammed up his ass (Rectum? Damn near killed 'im.) There is only one way to have that happen, and it is not by accident, even if you think you've managed to convince the ER staff that it was. And it is something that I would never, ever consider. As far as I'm concerned, that orifice is strictly Exit Only.

But I digress.

Yes, odds are you know someone with an alien abduction story. And if you don’t, you do now – because I have one of my own.

And yes, he explains this later:

Look, I’m not saying there isn’t life on other planets. Probabilistically speaking, there almost certainly is. But I don’t think any intelligent species has visited Earth, at least in any of our lifetimes. We have equipment sensitive enough to detect minor seismic activity on other planets. The amount of energy it would take to propel an object through space at a speed anywhere approaching practical would be so mind-bogglingly enormous we’d definitely know about it. Not only that, even assuming that aliens were visiting from Proxima Centauri, the nearest star system to ours, and even assuming they were traveling at speeds approaching c, we’d still see them coming for about four and a half years.

Minor nitpick: no, we wouldn't, because an object traveling at near the speed of light relative to us, headed in our direction, wouldn't... oh, hell, I don't have time to wrestle with general relativity right now. Point is, I still agree with his first points there, as I've banged on in here about way too often.

So here’s my theory, based on what happened to me. When I was a teenager, around fourteen or so, I started having night terrors. You don’t remember night terrors, but it didn’t take too long for my grandmother (whom I lived with) and my friends whose houses I would stay the night at to start waking me up in the dead of night, asking if I was okay.

Some of you lucky bastards don't remember night terrors.

I’d occasionally have nightmares I’d remember, and they’d always be variations of the same themes: I’d be lying down, completely immobilized. I wanted to move, but couldn’t. There was a blindingly bright light and I could hear what sounded like the cadence of speech, but couldn’t make sense of it. There was a looming figure I could vaguely sense, and then there was a blinding pain in my legs. Like hot razorblades being dragged along my calves.

Apart from the pain, which I fortunately haven't experienced, that sounds a lot like my own experience with sleep paralysis.

In other words, I had almost all of the signs of an alien abduction. But I know I wasn’t abducted by aliens. In fact, I know exactly what happened. See, when I was thirteen, I had a surgery on my legs because I was prone to walking on my tiptoes and it was giving me the backpain of a retired roofer.

And that might explain why my experience is different from his: I never had surgery as a child.

So, yes, I think that people have been waking up from general anesthetic during surgery, but remember it mostly subconsciously, like a dream. The bodily violation. The feelings of fear and immobility, like sleep paralysis. Confusion. Bright lights. Menacing creatures with bluish-gray skin, no nose (and sometimes no mouth), often with no hair, and strange, dark eyes? What, you mean like this? [picture of surgeon]

I think that's a fair hypothesis. It certainly doesn't explain every report of alien abduction, but it doesn't have to.

People probably start feeling all of this terror and strange dreams, and, unsure what’s causing it, latch onto the culturally ubiquitous idea of alien abduction. Remember when I mentioned that the Hill’s case, which set the pattern for the prototypical alien abduction story, took place in 1961? Well, in 1956, halothane was introduced as an inhalatory general anesthetic.

Sounds like a valid point. Of course, I wouldn't accept it as absolute truth until they do serious studies on it, but it's absolutely worth looking into.

Fentanyl, another drug that became used in general anesthesia, was first synthesized in 1960. All of this happened right around the time that there was an explosion in alien abduction claims.

Right now, we have possible correlation, but no evidence for causation. So, sure. I've been saying that alien abduction experiences track with sleep paralysis, but this works too. It could very well be both, or even have other explanations.

Like I said, though, don't think I'm just swallowing this whole. That would be just as rash as wholeheartedly believing in alien abduction itself. But it's a perspective that might have some merit.

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