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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Show Recent Comments

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.

December 4, 2021 at 12:01am

#1022750

It's A Conspiracy!

Another Cracked link today (I feel the need to warn people because I've heard reports that the site acts weirdly with some browsers, but I haven't experienced such a thing myself).

Anyway, the topic for today is conspiracy "theories."

8 Questions Internet Bozos Hate To Answer

I put "theories" in quotes because what most people think a theory is, isn't actually a theory. The common definition can be summed up as "guess." The scientific definition is a body of research that holds together and can be used to make predictions. The theory of gravity, for example, is remarkably good at predicting orbits and trajectories. Quantum theory is even more accurate. Neither of these are guesses; they're the result of centuries (gravity) or at least decades (quantum) of study, research, and experimentation.

But for the sake of my failing eyesight, I'll leave off the quotes from now on. We all know what a conspiracy theory is. I just wanted to emphasize that not a single one of them is a true theory in the scientific sense.

Now, I love hearing a good conspiracy theory. Who wouldn't want to hear about a secret Pentagon spaceship piloted exclusively by Sasquatches? (Short answer: they're leaving Earth because humans invented camera phones.)

Why would they do that when they already have the ability to appear horribly blurred in every photo taken of them?

And yet, I have a real problem with conspiracy theorists. For me, it’s sorta like Star Trek: love the stories, can’t stand the fan club.

I get this.

And just like any form of entertainment, I’ve noticed quite a few plot holes, and I have some questions.

Because this is Cracked, get ready for a countdown.

8. “Just … Why?”

Well, let’s throw that question right back at the conspiracy theorists: Why? What reasons do you have to believe these things?

The best scientists are a lot like children (only better educated and slightly less prone to painting the walls with pasta): they never stop asking "Why?" A parent might stop the endless barrage of whys by saying something like, "Because that's just the way it is." Scientists, if they have parents, don't listen to that answer.

Conspiracy theorists, however, have already decided the way things are and try to fit everything else to that.

Accepting the ugly truth might go against their most deep-seated beliefs, so instead they latch on to a comfortable lie.

And I get this, too. I don't think many people are entirely immune to it. For instance, I insist that I'm single because no one has figured out just how awesome I am, instead of me being fat, ugly, and sarcastic.

7. “Is This About You Being Right, Or Do You Just Wanna Make Other People Appear Wrong?”

The further down the internet rabbit hole these people go, the more they uncover supposedly privileged information the infamous “they” don’t want anyone to know about, and that leads them to believe that they now know more than even the world’s foremost experts on the subject. But worse than that, it also makes them cocky enough to pick a fight with everyone they encounter online. And now, we witness the Dunning-Kruger Effect evolving into a Debate-Me-Bro.

One of the core problems resulting from worldwide communication is that it's not as important to stand out in your own community, school, team, whatever -- you are now competing for attention with 7,899,999,999 other people (approximately). People want -- no, need -- to feel special. One way to do this in a crowded world is to decide you've figured out The Answer to something. Usually, you're wrong, but you'll never admit it.

And no, I'm not immune to this fallacy, either.

6. “OK … What Are You Selling?”

There is now a fourth reason conspiracy theories are so prevalent lately: It’s become a hugely profitable business model. The doomsayers used to be relegated to dark web bulletin boards and self-published manifestos, but now many of them have gone mainstream and built their own media empires. If it feels like conspiracy theories are getting crazier every day, it’s because they’re designed that way for a reason. Outrageous, inflammatory rhetoric generates clicks, and clicks boost ad revenue. Then the algorithms kick in and start suggesting similar content. Lather, rinse, repeat, and don’t forget to like, comment, share, subscribe, check out our Patreon, and buy our merch!

Even a cynic like me often falls into the trap of thinking that people are usually sincere with their statements. We forget that, sometimes, people are only sincere about one thing: wanting money.

This is kind of related to #7 above, the "standing out" part.

5. “You’ve Never Been In A Group Project, Have You?”

Every human being is fallible, and if you think you’re an exception to that rule … there’s your flaw. Regardless of your talent, your training, or your confidence, there is always a chance you could mess something up. When multiple people work together, one person’s strengths might balance out some of the others' weaknesses, but everyone involved brings with them their own unique potential to completely screw the pooch. That is true of any group project, be it in a classroom or the workplace, and if the tiniest speck of shit hits the fan, people will hear about it … but that strangely never seems to happen in a conspiracy theory, does it?

This brings up another issue that I don't think the writer covered. It's especially relevant to government conspiracy theories. The same people who are absolutely certain, in their own minds, that the government is capable of engineering vast projects (like, say, faking the moon landing) in near-total secrecy hold, simultaneously, the belief that everyone in the government is absolutely incompetent.

This is the pinnacle of cognitive dissonance.

4. “Couldn’t They Accomplish The Same Goal Without Being So Cartoonishly Evil?”

No one is gonna give a shit unless the stakes are ludicrously high and there are tons of A-list celebrities involved. If you were to tell people that there’s a cabal of Satanic child sex traffickers harvesting their victims’ adrenal glands to make psychedelic drugs, you’d get laughed out of that pitch meeting. But if you say Tom Hanks and Oprah Winfrey are attached to the project, that will get you some press.

That’s why theories like QAnon make no damn sense. They wouldn’t have to go to that much trouble. If Oprah really was a Satanist, she wouldn’t need to hide it. She could just select The Necronomicon for her book club, and two weeks later soccer moms everywhere would be sacrificing virgins.

I wrote a short story a while back about the scary influence Oprah has (or at least had; I don't keep up with that stuff). This is not a conspiracy theory; this is based on observational evidence. So yeah, drop in a celebrity or two and your idiotic ideas get spread around more widely.

3. “Does That Technology Even Exist?”

One of the weirdest tropes in conspiracy theories is when they involve the use of outrageous technology that even Roland Emmerich would think is lazy writing, and that guy’s next movie is about the moon falling.

I think that movie's already out by now. I wouldn't know. It's just about the stupidest thing involving the moon since the fake landing conspiracy theory. Speaking of which, the technology needed to fake that shit in 1969 would have been way more expensive and difficult to achieve than actually, you know, sending dudes to the moon. Not to mention the number of people involved; see above.

2. “How Do You Make A Peanut Butter And Jelly Sandwich?”

... actually, I'm afraid you're going to have to go to the link to read this one. It will make sense, I promise.

1. “What Happens If You’re Right?”

The one thing that all great conspiracy theories have in common is they all seem to be perpetually stuck in the middle act of the movie. The bad guys and their motivations have been established. We sorta have an idea of the magnitude of the threat we’re facing. If the story even has a hero, they're really not doing a great job saving the day so far. But let’s just assume the theories are correct and cut straight to the climactic showdown. Who do you think is gonna win?

Also this. Taking the government conspiracy theories as an example, if you're right and they're secretly engaging in mind-control, what exactly do you think is going to happen? Hell, if you're right, and you're spreading such a conspiracy, you've already disappeared, quietly, and they've put a robot or clone in your place to walk back the shit you said.

As the old saying goes, perhaps the greatest conspiracy is there is no conspiracy.

That said, there have definitely been conspiracies. Like Tuskegee

or the Roswell Incident

- which was a cover-up, though not for space aliens but for secret US technology. So yes, sometimes there really is a conspiracy.

But that doesn't mean that your pet theory holds any water.

Or maybe -- just maybe -- They seed these conspiracy theories themselves to hide the real truth, which is that we're being farmed by lizard people from Hollow Earth. If you throw enough falsehoods out there, the truth will be concealed like a wolf wearing wool..

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