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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Brandiwyn🎶

a day ago
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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 20, 2021 at 12:03am

#1019687

Tragedy

You might need to use a private window in your browser for this one, but not because of offensive content. I mean, maybe you'll find the content offensive; I don't know.

The Opposite of Toxic Positivity
“Tragic optimism” is the search for meaning during the inevitable tragedies of human existence, and is better for us than avoiding darkness and trying to “stay positive.”

I haven't mentioned much in here about toxic positivity, which is basically what used to be called Pollyanna syndrome: the idea that we should always maintain a positive mindset. To be clear, I call it bullshit, and apparently I'm not the only one.

In fact, I like to do just the opposite. Every silver lining has a cloud.

Refusing to look at life’s darkness and avoiding uncomfortable experiences can be detrimental to mental health. This “toxic positivity” is ultimately a denial of reality. Telling someone to “stay positive” in the middle of a global crisis is missing out on an opportunity for growth, not to mention likely to backfire and only make them feel worse.

Especially be on the lookout for any sentence that contains the phrase "at least." As in, "I know you just lost your arm in an accident, but at least you still have the other one."

As the gratitude researcher Robert Emmons of UC Davis writes, “To deny that life has its share of disappointments, frustrations, losses, hurts, setbacks, and sadness would be unrealistic and untenable. Life is suffering. No amount of positive thinking exercises will change this truth.”

1. "Gratitude researcher?" Fuck right off with that bullshit.
2. "Life is suffering?" Sod back off to Tibet with that crap.

The antidote to toxic positivity is “tragic optimism,” a phrase coined by the existential-humanistic psychologist and Holocaust survivor Viktor Frankl.

And right now I'm picturing an emo kid with a black t-shirt sporting an image of Frankl. (I had no idea what he looked like. Turns out: exactly as you'd expect.)

Tragic optimism involves the search for meaning amid the inevitable tragedies of human existence, something far more practical and realistic during these trying times.

Meaning? HA! Hang on while I put on my black t-shirt with the picture of Nietzsche. (I do know what he looked like. Epic mustache and all.)

The gratitude researcher Lilian Jans-Beken and existential positive psychologist Paul Wong created an “Existential Gratitude Scale” to measure the tendency people have to feel grateful for all of human existence, not just the positive aspects.

I'm starting to need booze for this crap. I seriously can't copy any more of this. The article, in my considered opinion, fails to make any meaningful distinction between "toxic positivity" and "tragic optimism." So I'm going to propose a third approach; there are probably many more.

What I propose is, as you might expect, drinkingcomedy.

There's a quote I used to use as the introduction to every Comedy newsletter I wrote. It's by Robert A. Heinlein, from Stranger in a Strange Land, and one version of it goes like this:

"I've found out why people laugh. They laugh because it hurts so much... because it's the only thing that'll make it stop hurting" ... "But that's not all people laugh at." "Isn't it? Perhaps I don't grok all its fullness yet. But find me something that really makes you laugh... a joke, or anything else- but something that gave you a a real belly laugh, not a smile. Then we'll see if there isn't a wrongness wasn't there."

I first read that book when I was a kid, which is probably one of the reasons why I'm so warped (along with the tunnel scene from Willy Wonka). The quote struck me as true at the time, and my life experience since then has only reinforced its veracity.

Think of a joke. Any joke; doesn't matter, as long as it's one that you absolutely laughed at (which means probably not jokes that rely on puns; those are mostly only painful to the victim of the telling of them). If you can't think of any, here's a joke that was once ruled to be The Funniest Joke in the World:

Two hunters are out in the woods when one of them collapses. He doesn't seem to be breathing and his eyes are glazed. The other guy whips out his phone and calls the emergency services. He gasps, "My friend is dead! What can I do?" The operator says, "Calm down. I can help. First, let's make sure he's dead." There is a silence; then a gun shot is heard. Back on the phone, the guy says, "OK, now what?"

Leaving aside for a moment the obvious problems with the delivery (not least of which is the use of passive voice), that joke absolutely relies on tragedy and misunderstanding. I mean, a guy is dead! Not just any guy, but the main character's friend! This would suck if it happened to you, wouldn't it? And the tragedy is compounded by the MC's misunderstanding of the instructions given, something that would surely keep him up at night for the rest of his life. But a whole lot of people laughed at the joke, as is evidenced by its status as World's Funniest Joke.

Of course, at this point that joke, and variations of it, have been circulating long enough that it's probably not as funny as it was when it was fresh. Also any time a joke is analyzed like I just did, it becomes not funny. And finally, senses of humor differ; you personally may not find it amusing. So here's another one with less death involved. It's from the same project to find the funniest joke:

Sherlock Holmes and Dr Watson were going camping. They pitched their tent under the stars and went to sleep. Sometime in the middle of the night Holmes woke Watson up and said: “Watson, look up at the stars, and tell me what you see.”
Watson replied: “I see millions and millions of stars.”
Holmes said: “and what do you deduce from that?”
Watson replied: “Well, if there are millions of stars, and if even a few of those have planets, it’s quite likely there are some planets like earth out there. And if there are a few planets like earth out there, there might also be life.”
And Holmes said: “Watson, you idiot, it means that somebody stole our tent.”

I'm not going to ruin that one by analysis; hopefully the essential tragedy at the core of the joke is obvious. Yes, comedy also requires the unexpected, but there's also, almost always, an element of what in another context would be tragedy or suffering.

I know I've said this sort of thing before, even quoting the same jokes, probably in a Comedy newsletter, but it's been a while and it's relevant to my thesis today. My point being that we all know that life can sometimes suck, and one of the main purposes of humor is to acknowledge this and prepare for it. While it's hard, perhaps even impossible, to find humor when you, personally, have just received bad news, comedy can still work to take the edge off if it's not related to the subject at hand. As Mel Brooks once noted, "Tragedy is when I cut my finger. Comedy is when you fall into an open sewer and die."

And bad shit will happen to all of us. As I've said before, my eyesight is deteriorating, and I'm going to have to have cataract surgery. I have a phobia about anything touching my eyeballs (though my eyelids get a pass there). How do I deal with this? I say things like, "Damn, I shouldn't have watched so much porn."

There is (quite literally) no "bright side" to losing one's vision. But humor takes some of the sting out of it. Now if you'll excuse me, I have to go shave my palms.

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