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 10 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 12 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.

May 1, 2022 at 12:01am

#1031744

The Birds

There is no way to disprove a conspiracy theory.

‘The lunacy is getting more intense’: how Birds Aren’t Real took on the conspiracy theorists
On a march, Peter McIndoe held up a sign and talked about how the ‘deep state’ had replaced all birds with drones. It was meant as a small act of satire but has become a mass movement

What I mean by that is, of course you can disprove a conspiracy theory. What you can't do is present any evidence that would make someone who's invested in it change their mind.

Either they'll dismiss it as fake, or declare that you're obviously part of the conspiracy.

This seems to be a defining feature of humanity; we can believe any bullshit if we really want to, and once we do, it becomes part of our identity, and to break with it can cause tremendous mental anguish.

I say "we" because while I try to be open to new evidence, I'm still nominally human, and I can't say I'm totally immune. Part of the problem is when we get deep into these things, we don't even recognize our own cognitive issues. Like, I'm convinced, with absolutely no evidence, that PETA is secretly run by soybean growers and fake fur manufacturers, both of which have a massive financial stake in getting people to eat no meat and wear no fur. I'm not sure what could cause me to believe otherwise. But really, does that particular conspiracy theory do any harm?

(The above link is from The Guardian, so some of these quotes will have British-spelled words.)

Anyway. One small act of satire:

McIndoe made a placard, and went out to join the march. “It’s not like I sat down and thought I’m going to make a satire. I just thought: ‘I should write a sign that has nothing to do with what is going on.’ An absurdist statement to bring to the equation.”

I don't know if I've mentioned in here before the power that I think absurdity has. Sometimes you have to venture into the surreal to truly comprehend reality.

That statement was “birds aren’t real”. As he stood with the counterprotesters, and they asked what his sign meant, he improvised. He said he was part of a movement that had been around for 50 years, and was originally started to save American birds, but had failed. The “deep state” had destroyed them all, and replaced them with surveillance drones. Every bird you see is actually a tiny feathered robot watching you.

Obviously false. Hell, I ate some chicken just a few hours ago (suck it, PETA), and a chicken is a bird.

Or... wait... WAS it chicken? How do I really know?

Anyway, the "theory" is obviously false. After all, birds are tough to fake. Insects and spiders, now... those are actually government drones spying on you.

It’s a made-up conspiracy theory that is just realistic enough, as conspiracies go, to convince QAnon supporters that birds aren’t real, but has just enough satirical flags that generation Z recognises immediately what is going on.

Right, because every generation is a solid cohort that moves in lockstep. No Gen-Z doesn't recognize it as satire. No Millenial, Xer, or Boomer recognizes it as satire. We are all one. Conform.

It’s a vivid dramatisation of how divisive conspiracy theories are; people who believe them live in another world, where any wild theory flies and even the most fleeting attempt to fact check it or test it against logic (if birds have been destroyed, who’s eating all the worms?) marks you out as a brainwashed liberal.

The Guardian leans left, and we all know it. I dislike playing the "both sides are bad" card, but here again, as with the generations thing, let's not forget that many liberals subscribe to conspiracy theories, too. Like my pet PETA one.

The paedophile element of QAnon, where Hillary Clinton and a huge global web of powerful liberals, are abusing children and keeping them in tunnels, sounds completely unhinged.

Not if you understand that for those people, every accusation is actually a confession.

They’re unified on the prank, right? There aren’t people there who think birds genuinely aren’t real? (I still need a lot of footnotes.) “Yeah, they’re role-playing together. They’re role-playing the collective understanding of the conspiracy theory.”

Yeah, I wouldn't be so certain of that. I'm sure there are people cosplaying as flat-Earthers, but I'm pretty sure a lot of the flat-Earthers believe every word they spout.

Anyway, the article is a nice glimpse into a conspiracy theory that we know is satire, kind of like we know that the Flying Spaghetti Monster is a satire of religious nutjobs.

1 Like • Comment • View 2 comments

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