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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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

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

October 18, 2019 at 12:20am

#968052

On Love and Biochemistry

Another Cracked article today. This one's from David Wong, who usually has some good perspectives on things.

https://www.cracked.com/blog/how-our-pets-have-evolved-to-emotionally-manipulate...

How Our Pets Have Evolved To Emotionally Manipulate Us

To be fair, we, too, have evolved to emotionally manipulate us. Don't believe me? Okay, what do you do when you hear a crying baby? (Well, I get as far away from it as possible until it shuts the hell up, but I'm told that actual human beings try to comfort it.)

It's easier to love a dog than it is to love a spider or a cactus because dogs have adorable facial expressions. The nonverbal pleas for food and love, the dopey smiles, the heartbreaking, almost teary disappointment -- it's all there. But the reason they can make those expressions is that, after evolution split dogs from wolves to begin their partnership with humans, they evolved facial muscles to make "expressions" that humans would respond to. They grew little eyebrows.

It is true that, for most people, our love for something is proportional to its cuteness. Sure, you can convince yourself that a snake is adorable, but mostly that means you're trying to be metal.

"But cats don't have those expressions, and we love them just the same!" you may object. "If anything, cats look like they hate our guts!" Sure, but they just found another way to rope us in. Did you know that cats don't meow to each other? It's a sound they've developed specifically for humans, to get us to feed them. Even creepier, it may be specifically calculated to mimic a crying human baby. If anything, they think it's hilarious that we keep falling for it.

So remember a couple paragraphs ago, I mentioned that human baby screeches get on my very last nerve? I wasn't kidding (pun intended, as always). So, one time, I was in an airport - a setting guaranteed to enhance any slight annoyance and/or frustration to my breaking point - and I heard two things in close succession. First, a baby wailing; I noped right on out of there. Second, someone had a cat in a carrier and it emitted that frightened yowling wail that only cats can manufacture - and all I wanted to do was reach in and comfort it, regardless of any scratching it might do if I tried.

Consequently, I'm not sold on the "cat cries mimic human baby blurbling" theory.

This would, of course, mean that it's entirely possible to pour all of your love into something that is in fact coldly sending back just enough validation signals to keep you doling out the food and shelter. It would open the possibility that the kind of sweeping gooey sentimentality we think of as the Most Important Thing In Life will be seen by future societies as a weird, backward superstition. "These sick bastards went so far off the rails that they literally defied their most basic biological imperative and adopted pets instead of having children."

Yeah, look, I know a lot of people refer to pets as their "children," but again - I always wanted pets; I never wanted children (at least since I was old enough to physically help to produce them). My cats are my cats, not my "furbabies" or child-surrogates or whatever. Life isn't binary - it's not "kids, including surrogates" versus "a life of utter solitude." Anyway, point is, emotion is a biochemical process, and chemistry doesn't care.

What if all of our poetry and lyrics about "soul mates" and eternal devotion is one big weird cultural fetish?

What if water is wet and grass is green? Oh, but this author thinks I'm being cynical. I'll show you cynical, Wong.

But the gooey sentimentality, it turns out, is just as easy to defend with cold logic. If love isn't real, then nothing is. "But it's just an idea!" So's the law, and democracy, and money. If you can argue away love as just hormones and brain chemicals, then you can argue away the sun as being just a bunch of hydrogen and helium molecules. Figuring out how a thing works doesn't invalidate its existence.

But... the sun is a bunch of (primarily) hydrogen and helium molecules, and its goal is to give us cancer. I'm not saying love isn't real; I'm saying it's an emergent property of consciousness, which itself is an emergent property of biology, which is essentially a long, sustained chemical reaction. But as with everything else, we can make of that whatever we will.

Being willing to believe in things, to buy in, to put yourself out there, that's what takes real courage.

Sure, Wong, and I believe I'll have another beer.

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