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

March 25, 2021 at 12:01am

#1007019

Certain Things

Not only is this one ancient now (2014), but it's been hanging out in my queue like panhandlers outside of McDonald's for several months now.

Planning Your Future Is Pointless. The How And Why Of Embracing Uncertainty
Wondering what the future holds is a tough question at any age. Instead of trying to figure it all out, get comfortable with the discomfort of uncertainty.

Look, I'm all for embracing uncertainty, but "planning is pointless?" Clickbait? Or something worse?

Life doesn’t go according to plan, and while a few people might do exactly what they set out to do, you never know if you’re one of those. Other things come along to change you, to change your opportunities, to change the world.

I've found the trick lies in retconning your ambitions. Like, say you're a teenager and want to be a doctor. You end up running a backhoe at the local landfill (nothing wrong with that; it's just about as far from "doctor" as I could think of). So you retcon Teenage You into someone who always wanted to run a backhoe.

Hey, it worked for me; I never wanted to be an astronaut.

So if you can’t figure out the future, what do you do? Don’t focus on the future. Focus on what you can do right now that will be good no matter what the future brings. Make stuff. Build stuff. Learn skills. Go on adventures. Make friends. These things will help in any future.

Nothing wrong with any of those activities, but screw "don't think about the future."

One of the most important skills you can develop is being okay with some discomfort. The best things in life are often hard, and if you shy away from difficulty and discomfort, you’ll miss out.

On the contrary, I find that the best things in life are free, easy, and instantly gratifying. If you work hard, something will come along to change something in yourself or your environment, and all your hard work won't have paid off. Still gotta think about the future, though.

Learning is hard. Building something great is hard. Writing a book is hard. A marriage is hard. Running an ultramarathon is hard. All are amazing.

I've never run an ultramarathon, fortunately, but I've done those other things. I wouldn't call any of them "amazing."

Try writing a blog or meditating every day.

Um, hello.

If you’re good at discomfort and uncertainty, you could do all kinds of things: travel the world and live cheaply while blogging about it, write a book, start a business, live in a foreign country and teach English, learn to program and create your own software, take a job with a startup, create an online magazine with other good young writers, and much more.

You can also do all those things if you're independently wealthy. I don't want to travel cheaply, though. I want to experience the finer things, like great wine and five-star hotels.

All of this is useless if you can’t overcome the universal problems of distraction and procrastination.

Welp, I'm boned.

Learn about how your mind works, and you’ll be much better at all of this. The best ways: meditation and blogging. With meditation... you watch your mind jumping around, running from discomfort, rationalizing. With blogging, you are forced to reflect on what you’ve been doing in life and what you’ve learned from it.

Nah, it's more fun to blog about peoples' deadline-driven articles. As for meditation, I've tried it; I fall asleep every time.

Speaking of which, have you ever had -- I don't know what it's called, but the exact polar opposite of a nightmare? Like, in a nightmare, you wake up sweating, heart racing, maybe even crying out in terror (or in my case, being unable to cry out in terror because no voluntary muscles work). With what I'm talking about, you wake up at utter peace with yourself and the world. And get your mind out of the gutter; there was nothing sexual about the dream, though I don't much remember what it was about. Mostly I just remember the feeling of great well-being, the opposite of the fear that a nightmare engenders.

It's happened to me a few times, most recently just two nights ago. Since there's not a name for it as far as I know, I have no idea if others experience that sort of thing.

I don’t think money is that important, but making money is difficult.

On the contrary, I think money is incredibly important. Enough money and you can basically tell everyone who's trying to tell you what to do to fuck off. You can set your own schedule. You don't have to worry so much about where your next meal is coming from. You don't have to spend time and energy on trying to find the cheapest everything (though it can be fun to do so). Most of all, if you get sick, you can actually get medical treatment (this last bit only applies in backwards countries). That old canard about how money can't buy happiness? Maybe, but it can buy beer, and that's close enough for me. It brings its own set of problems, of course, but they're better problems to have than "oh shit I can't afford this year's rent increase."

Protip: save an emergency fund, then start investing your earnings in an index fund and watch it grow over your lifetime.

Oh, look, one gold nugget amidst all the pyrite.

So, I don't know. The article is obviously aimed at someone much younger than me, and the world has moved on since he and I were starting out. As hard as things were in the 80s, they're exponentially tougher now. I think the basic advice has some merit, but not everyone's cut out for a life of pure uncertainty.

And yet, there's enough uncertainty that we should definitely plan for it.

Good luck with that.

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