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 14 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 16 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.

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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').

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.

April 24, 2022 at 12:04am

#1031265

Mothman

Y'all might have figured out by now that I have a thing for cryptid legends.

The Urban Legend Of The Mothman, West Virginia's Famous Red-Eyed Omen

Come to think of it, I wonder why there isn't a cryptid associated with the Great Dismal Swamp (from yesterday's entry). I guess because the reality was fucked-up enough that they didn't need to invent a... I dunno... bearigator or something.

I swear if I find stuff about a bearigator all over the internet after this, I'mma get angry. Unless they credit me for what is, really, come on, an awesome idea.

There are no alligators in the GDS, by the way. Not yet, anyway, not that they know of. I've seen them a bit southeast of there, though, around the Outer Banks of North Carolina, so what with climate change and all, it's probably only a matter of time.

Anyway. Today we're talking about our neighbor to the northwest.

If there’s two things the state of West Virginia is known for, it’s an unfortunate dependence on the dying industry of coal mining, and the legend of the Mothman.

That's a joke, right? It is Cracked, so I'm assuming it's a joke. If there's two things West Virginia is known for here in Real Virginia, it's strippers and hillbillies. If there's three, it's also stripper hillbillies. I'm not saying I've gone to hillbilly stripper clubs in West Virginia, so don't ask me how I know this.

More seriously, though, don't believe all the stereotypes. It's a beautiful state and the part they're talking about here, Point Pleasant, is a rare case of a town actually being named something descriptive (see also: Lynchburg, VA).

The first sighting of the beast comes from November 1966, in Clendenin, West Virginia. The first people to spot him were a group of gravediggers, which seems like a great job to pick up if you want to be creeped out all the time.

Ever notice that in movies and shows, no one ever digs up a grave during the day? It's always at night. That way no one's surprised when zombies attack.

The gravediggers described a huge flying figure dashing from tree to tree. They further described its appearance as a “brown human being,” which sounds like something a nervous neoliberal mayoral candidate would say while campaigning in a black neighborhood.

Another thing WV is known for is rampant racism, but I don't think this is a case of that. It's more a case of... cryptidism?

The article goes on to quote the newspaper account of the Mothman's second sighting:

"Two Point Pleasant couples said today they encountered a man-sized, bird-like creature in the TNT area about midnight last night. Sheriff's deputies and City Police went to the scene about 2 o'clock this morning but were unable to spot anything. But the two young men telling their story this morning were dead serious, and asserted they hadn't been drinking."

Suuuuuuure they hadn't.

However, he also said that it looked like “maybe what you would visualize as an angel,” which is like, the number one thing not to say if you’re trying to avoid a rubber room.

Yeah, I don't know about that. People claim to see angels fairly often. It's when their opposites, demons, start whispering in the person's ear that they get fitted for a nice canvas jacket with straps.

The legend continued to grow over the years, but one thing that never received further clarity was the Mothman’s motives, or relationships with the population he was allegedly haunting. There are no reports of direct violence or aggression.

You want to find out his motives? Ask him. I'll be right over there.

Perhaps the most unsettling sighting was the claimed sighting of the Mothman standing on the local Silver Bridge, which would collapse the next day, killing over 40. This served to add a new supernatural level of mystique to the Mothman, that of an omen, or for the especially suspicious, a saboteur.

I should note here that I don't know where the Silver Bridge was, or if it's been rebuilt, but the whole point of Point Pleasant is that it's located at the confluence of the Ohio and Kanawha Rivers. So there are a good number of bridges in the area. Some of them have even stayed up.

Oh, and yes, I've been there. There was, in the Before Time, a brewery pretty close to the Mothman statue. They brewed, among other things, hemp beer. It tasted like Mothman piss. The statue, by the way, is illustrated in the link. It is GLORIOUS.

I talked about the brewery before, about a year ago: "Liberating the Angel"

Some researchers note that this appearance is very common in the description of sleep paralysis demons, the sightings or hallucinations of which can stick in the subconscious and manifest later during times of fear.

I've never actually seen -- or, I guess I should say, I never hallucinated -- a sleep paralysis demon, despite my struggles with the affliction. Usually it's an inchoate presence. Sometimes there are footsteps, which you wouldn't expect from a flying cryptid.

The flying is also not a particularly unique detail in a location with plentiful wildlife, specifically occupied by populations of herons, which are, well… huge, man-sized birds. From this comes one of the most compelling scientific theories of the Mothman. Dr. Robert L. Smith postulates that the original Mothman, the one seen frequently and in detail in the 1960s, was in fact a sandhill crane.

Yeah... I looked up "sandhill crane," and a) they're not human-sized and b) they don't live in the Ohio River Valley. I'm all for finding logical explanations for these sorts of things, but I'm not convinced about this one. On the other hand, it's sometimes hard to tell actual size at night, and birds sometimes fly outside their established ranges. So, okay, maybe.

When you take all this into account, it seems very likely that the area was occupied by an unusually large sandhill crane, perhaps even one that did possess some level of mutation caused by chemical leaks.

Common enough in the Ohio River Valley.

I will point out, though, that genetic mutation doesn't work that way. You don't expose a crane to gamma rays and thereby turn it into Heron Hulk.

The lifespan of a sandhill crane is around 20 years, and one could assume even less if the bird was indeed mutated or sickened by chemical leakage, which would explain its fairly brief reign of terror. Which is to say, the Mothman was probably dead long before you were born.

Except the first sightings were during the year I was born. Not all your readers are whippersnappers, Cracked.

Anyway, the reason I'm fascinated by cryptids is not that I "believe" in them; it's because of what the legends tell us about ourselves. Was there an actual Mothman? Hell no. But I accept that people saw something, and part of the fun of life is that we can't definitively explain everything.

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