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').

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.

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

1 Like

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

March 23, 2023 at 10:35am

#1046875

Unnatural

I've said before that "natural" as a food or other product label is bullshit.

5 ‘All-Natural’ Products That Are Complete and Utter Garbage

Good to know Cracked has my back.

You don’t have to be an Amish man, churning butter and spitting distastefully at the idea of an electric can opener, to admit that the grip tech has on our lives is a pretty tight one.

I like it that way. Could do with less surveillance, but other than that, bring on the convenience.

So what is there to do? What humans do best: violently overcompensate for something that upsets them by swinging back in the other direction.

See also: tiny houses as a reaction to McMansions; "cluttercore" as a perfectly reasonable response to Marie Kondo.

This sort of panic pops up in all sorts of ways, from going back to flip phones, to deciding to eat like cavemen (despite the fact they were never known for their health).

Health, no. Probably stronger than fuck, though. But that wasn't because of what they ate, but how they caught it.

As such, there’s been a massive movement of “natural” products that have come along to capitalize on people’s distrust of words that start with “di-” and end in “-ate”, and, looking at their price tags, I do mean capitalize.

Everything you eat or drink is made of chemicals.

So, in usual fashion, five items on a countdown list.

5. Natural Peanut Butter

A jar of classic Jif or Skippy is an absolute delight in every possible way. Cutting into that perfect plateau atop a new jar’s contents, watching it curl and collect along the blade of a knife and then spreading it like sweet mortar onto your delivery method of choice.

Dammit, now I'm hungry. Goddamn porn writers...

I guess this one's a matter of opinion. I kind of like the "natural" labeled kind. I prefer almond butter, though.

4. Natural Deodorant

Natural deodorant. That old borderline oxymoron. Here’s a tip: If anyone ever asks if you’re wearing natural deodorant, it means you smell like an upside-down porta potty.

This. Once we were thoroughly sold on the idea that we should smell like soap and not like human, there was really no going back.

For most of us though, the reason given for using natural deodorant is mostly avoiding aluminum, for antiquated reasons that plenty of scientists now say aren’t proven.

I thought aluminum was just used in anti-perspirants, which I know come bundled with deodorants, but you can get the kind of the latter that's not also the former.

3. Oil Pulling

I first heard this referenced maybe a year or so ago? I don't know; my time sense is warped. And my mind went, "bullshit."

It comes from ancient medicine dating back to hundreds of years B.C. You know, back when they had medicine super figured out, and everyone was the epitome of health. Somehow, it’s come roaring back, mostly thanks to the kind of people who spend $200 a month on crystals because dying is scary. Unsurprisingly, the general medical consensus on the practice is, “Look, if you want to dump a bunch of oil in your mouth, we’re not going to stop you, but also, this is probably bullshit.”

Yeah, maybe they gave up on it because it wasn't doing squat.

Doubt it's harmful, though, so whatever.

2. Hair-Growth Cures

There are only two things proven to slow or reverse hair loss, and neither of them grow on trees: They’re the medications finasteride and minoxidil, and even they aren’t guaranteed. If you’re wondering if some new natural, simple hair regrowth solution works, the answer’s right in front of you, in the form of a world full of balding men that sure wouldn’t like to be.

It's questionable that one can consider most hair loss to be something that must be "cured." Want to be "all-natural?" Accept male pattern baldness; it's called that because it's genetic in origin. Now, I understand wanting to not be bald. But I also understand leaning into it like Jean-Luc Picard.

1. Vitamin C

I know, I know. I’m sorry to come for your precious packet of no-sick powder, especially in a time of public health panic. But mega doses of vitamin C, in whatever form you ingest them, have been proven time and again to be psychosomatic more than scientifically helpful.

Funny thing about the placebo effect (which is psychosomatic): if it works, why not take advantage of it?

I'd always heard that you can't overdose on Vitamin C, that excess isn't bad for you. And that's probably true in doses that won't explode you first, though there is a such thing as too much of anything. But what is bad for you is spending a bunch of money on something that doesn't work as advertised, especially when you need to save that so you can pay at least part of your inevitable medical bills.

In the end, "all-natural" is a marketing strategy, nothing more. Plenty of natural things will make you sick or otherwise mess you up, like poison ivy or the mushrooms I've talked about in here before. While there are certainly human-developed chemicals that are dangerous, too, having a knee-jerk reaction to all of the stuff they put in food is probably taking things a bit too far.

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