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 15 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.

April 18, 2022 at 12:01am

#1030917

Legendaries

Just so we're clear, all urban legends were made up by someone. By definition. Sometimes that person is otherwise known for other things, though.

6 Urban Legends That Were Totally Made Up By Celebrities

By "celebrities" here, Cracked isn't talking about, like, Brad Pitt or Kim Kardashian, but some well-known historical figures.

Celebrities: They're just like us, except people actually care what they have to say.

Ah, but which is chicken and which, egg?

Famous people have been accidentally etching bullshit into the public consciousness since, well, as long as famous people have existed.

"Accidentally" gives some of them too much credit.

6. The Jersey Devil Myth Was Started By Ben Franklin

The Jersey Devil is kind of a less-famous Bigfoot -- a humanoid reptilian cryptid with a fearsome appearance that lives in the Pine Barrens of South Jersey.

There is (or was; there's been a business-shattering pandemic since I went there) a brewery in New Jersey called River Horse. You might recognize River Horse as the literal translation of "hippopotamus." So their mascot is a hippo. But on one of their beers, that hippo is drawn to look a lot like the Jersey Devil. If, of course, there were such a thing as a Jersey Devil.

I mention this because I think Ben would have been amused, too.

As Cracked has previously mentioned, Franklin had a lively rivalry with a local New Jersey politician and publisher, Titan Leeds. Leeds made for quite the easy target, as he was a proponent of astrology and backed the unpopular local governor. In his usual "ain't I a stinker" way, Franklin attempted to smear Leeds by writing a joke article claiming a monster was born to the Leeds family. It wasn't the classiest display, and in an unfortunate coincidence, the Leeds family had some disabled members, and one was born in the same year as that eventually attributed to the Jersey Devil.

And that, kids, is also how trolling was invented.

Franklin, through the magic of his words, summoned a powerful force that, once unleashed, cannot be killed. That force is called Bullshit, and its dark powers are not to be trifled with.

Truth.

5. The Hollow Earth Theory Was Invented By Edmond Halley (Of Halley's Comet Fame)

I cut Cracked some slack because it's not pretending to be The New Yorker, but calling some things a "theory" is an affront to actual theories.

Still, the theory persists to this day among people who have no idea how planets work.

Which, the more I see of the internet, is a whole hell of a lot of people.

This one came from Edmond Halley. The comet guy. The year was 1692, and Halley was an astronomer who liked to pal around with Isaac Newton.

Halley did a shitload more than observe the comet that would end up named after him and eventually bookend the life of Mark Twain. But no, let's call him the "comet guy." That's a bit like calling Einstein "the photoelectric effect guy."

In order to explain variations and inconsistencies in compass readings, Halley proposed that the Earth we lived on was just an outer shell, and that there were one or more concentric layers of inner-Earths surrounding a central core Earth, each separated by their own atmospheres.

I mean, it's not terrible science to put forth a hypothesis, however outlandish it may seem to modern eyes. Wouldn't have taken much to disprove, sure, but weirder explanations have been proposed for shit we didn't understand until we did.

Oh, and there was also maybe an advanced civilization within the Earth, glowing lights, and seeping gas that created the aurora borealis. That too.

It seems to be human nature to believe that since we live in civilizations, other life-forms must also create civilizations. Hell, Star Trek is pretty much based around that idea. As yet, there is no evidence that evolution must produce civilization-developing species; only that it can.

Four centuries and a few hopeful scientists and explorers later, lots of educated people still buy this hilariously stupid theory. Well, shit, now we're wondering if that comet is real.

Well, I've never seen it. And never will, unless by some quirk of fate I live into my 90s and can still see.

4. The New York Sewer Alligators Were Popularized By Best-Selling Novelist Thomas Pynchon

Gravity's Reptile?

The story goes that New Yorkers of the early to mid-1900s brought back baby alligators from Florida vacations, only to be dismayed at the animals' striking increase in size. What to do? Flush them down the terlet, of course!

I remember some kid repeating this one in middle school, very seriously. I don't know whether he bought into it, or if he was messing with me. Probably both. (I was known to take trips to New York even then, with my family.)

To be fair, he didn't invent the ridiculous myth out of whole cloth -- he just cemented it in the public consciousness.

As with Ben Franklin, someone like that writes something, someone's bound to take it seriously.

A generation earlier, in 1935, the New York Times reported that a single 125-pound alligator had been found in the sewers, but was killed because it turned savage.

Okay, well, I'm no herpetologist (never had herpes), but from what I understand about gators, they don't "turn" savage; they "are" savage. At least from an anthropomorphic perspective.

3. Charles Dickens and Oscar Wilde Convinced The World The Chinese Loved Opium Dens

They're always shadowy places run by cunning Asian stereotypes who deal in organized crime and every depravity known to man. The floor/furniture is littered with passed-out addicts. And, as you might have guessed, they're almost entirely a fictional invention.

Well. I suppose it's good to know that drug panics and racism have gone hand-in-hand for at least a couple of centuries. I said "good to know," not "good."

The irony is that most Londoners at the time would have been able to buy opium at their local chemist in the form of over-the-counter cough syrups -- the substance wasn't destroying the East End, it was destroying the irritation in their sore li'l throats. But forget about facts, there's racial fear-mongering to be had!

At least some of the modern versions of this -- meth labs, Purdue Pharma, etc. -- expanded the moral panic to other races.

2. The Mystery Of The Marie Celeste Was Created By Sir Arthur Conan Doyle

Which I was only introduced to by an early (relatively speaking) Doctor Who episode. I think it was the Fourth Doctor, which would put it in the 1970s. And if I recall correctly, they solved the mystery: it was Daleks! The mystery of how those rolling pepper pots were able to board and exterminate all the sailors on a ship with multiple decks wasn't addressed until the series reboot in the noughties.

Like the authors who would follow, he grabbed attention by advancing a particularly salacious theory about what went down: The ship's abandonment was caused by a black passenger staging an uprising against the white crew. He hijacked the ship, sailed it to Africa, and murdered everybody. Yes, Doyle knew that pushing those racial buttons would sell.

Sigh.

He went on to create Sherlock Holmes a few short years later, a character obsessed with sorting through bullshit to find the truth. Feeling guilty about something, Mr. Doyle?

To be (somewhat) fair, it was pretty much standard practice in pre-modern fiction writing to pretend that you're telling a true story, and even claim it to be fact, not fiction. This was generally understood, as far as I know, to be a literary device. So I don't know if Doyle could have expected his writing to be taken at face value -- though, apparently, he wasn't too upset that it was.

1. Virginia Woolf Invented The Puritanical Victorians

But, as Cracked has mentioned before, the idea of the Victorians as prudes isn't borne out by the facts. This perception was largely started by a group of intellectuals known as the Bloomsbury Group, which included the famous economist John Maynard Keynes, writer E.M. Forster, and the legendary wordsmith Virginia Woolf. They were cool.

"They were cool." Honestly, could anyone wish for a more beautiful epitaph?

It's not that the Victorians were stuffy and sexless; it's that every young creative thinks their parents are stuffy and sexless. This too-cool-for-old-school clique's most famous member was Woolf, who attacked the values of her elders through her writing and took pride in her generation's supposed rejection of them. Like so many before them, the Bloomsbury Group sought to bill themselves as progressive by knocking the olds.

OK Boomer.

It's like I've been saying: today's generational split is but a rhyme of all of those that have gone before.

So there you go: If you want to create a story that will withstand the test of time, first and foremost it must be something that people really want to believe.

Which, after all, explains a lot.

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