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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Turkey DrumStik

about 2 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

a day 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

a day 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').

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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

February 17, 2019 at 12:28am

#952190

Lone Asshole Theory

https://thewalrus.ca/where-not-to-travel-in-2019-or-ever/

“My name is John!” shouted John Allen Chau from his kayak in November 2018 as he paddled toward strangers on the beach of North Sentinel Island in the Bay of Bengal. “I love you and Jesus loves you!” In response, the people on the remote Indian island strung arrows in their bows.

Yeah, I heard about this asshat when it happened. Got what he deserved.

Unfazed by the genocidal threat his germs posed and fresh out of missionary boot camp, Chau made repeated attempts to land—ignoring arrows and Indian law—in an effort to bring the Gospel to the Sentinelese.

"I know better than anyone else; the only law that applies to me is God's Law; and I'm doing The Right Thing!" Lord, save us from your followers.

Media coverage of Chau’s acts was disturbing because it didn’t come off as coverage of a crime—at least, not of his crime. Other major news outlets similarly valorized Chau’s legally and morally corrupt foray, highlighting his conviction as if tone-deaf temerity were a quality to admire.

Yeah, that bugged me at the time, too.

So, basically, his act of trespassing was lauded by exactly the same kind of people who are perfectly happy to put up "Trespassers will be shot" signs and point shotguns at census workers. Yeah, that's consistent.

As someone who has been called an adventurer before, I feel more of a sense of kinship with the person on Twitter who suggested this fix for the Times headline: “Remote Community Faces Biological Terror Threat From U.S. Religious Extremist Killed by Local Authorities.” To extol or glamorize any aspect of what Chau did risks condoning a brand of colonialism that should be anachronistic by now, and not just among missionaries. In fact, Chau’s evangelism is too easy a target, and it’s one that eclipses his more fundamental transgression.

Pretty much this - though there's no way to know if the Sentinelese would have been aware of the bioterror threat he represented.

This sort of thing puts me in mind of Star Trek. Okay, bear with me here.

I seriously doubt we live in a Star Trek - type universe, with sentient alien races around every other star. The evidence just isn't there to support that idea. In fact, it's possible that we're the only intelligent life in the galaxy (not the universe, of course, but the galaxy). And before you start with the "no intelligent life" jokes, I'm using the term in the sense of "technologically advanced enough as a species to go into space." We meet that definition. We know of no other species that does. Also, I'm not saying that life isn't out there; just that there's absolutely no law that says intelligence (that definition, again) must evolve in an ecosystem, and that the circumstances that allow it might be exceedingly rare. Like, 1 in 10^100 rare.

Okay, I digress, but it's important to my point. Suppose I'm wrong. Unlike that missionary dickwad, I can allow for the possibility. How do we keep from fucking up first contact?

Star Trek proposed a sort of legalized, formalized ethical framework in the form of the Prime Directive. You know it, I'm sure, but basically it's "don't mess with pre-warp cultures." Various starship captains then proceeded to find reasons to bend or break it, but that's not important right now; what's important is that it's codified ethics. I have my issues with it as a concept - like Asimov's Three Laws of Robotics, it seems hopelessly naive, for one thing - but it can be the basis for a discussion.

Thing is, we can go ahead and discuss it all we want, but there's another problem. It probably has a formal name, but I have no idea how to look it up. I'll call it Lone Asshole Theory, or Waltz's Fourth Law: It doesn't matter how ethically 99.999% of us act, if the actions of the other 0.001% can screw things up for everybody.

For example: Most of us wouldn't steal from a car, even if it were unlocked with the windows down. But we roll up the windows, lock the car, and remove valuables because some people do that. Most of us wouldn't set off a nuke in a major city, but all it would take would be one lone asshole to ruin everybody's day. The vast majority of humans wouldn't have slammed airplanes into skyscrapers, but those few that did made life more difficult for everyone else.

Similarly, we could have a bunch of protocols in place in case we ever met aliens, but everything would be completely undone by one John Allen Chau. All it would take would be one inadvertent biological contaminant (unlikely, of course, but can't rule it out), or one twat on a "mission from God" to convert the Little Green Aliens to Christianity. Or whatever religion. Hell, there'll probably be missionaries from a whole lot of religions falling all over themselves trying to spread the "good" news. I call this a nightmare scenario.

Waltz's Fourth Law basically explains why we can never have nice things. It's akin to the Tragedy of the Commons, but in that scenario, there are usually several bad actors; in mine, all it takes is one.

So, do I have a solution? Hell, no; of course not. And like I said, it'll probably never come up. Almost certainly not in my lifetime. So the only thing I can think of is to continue to make people aware of the problem, and to do what we can to stop the Lone Assholes.

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