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.

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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.

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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.

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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.

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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.

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January 31, 2021 at 12:08am

#1003296

Super Genius

I considered skipping today.

As of right now, my daily blogging streak exceeds 13 months. At some point I will break it, but the later I skip a day, the more I'm likely to plunge into existential despair when it happens (even if it's on purpose).

But then I looked at my Blog Fodder collection, where I store some of the interesting links I come up with in my perusal of our version of the Library at Alexandria. The past three months have been entirely prompt-oriented, and while it's been fun and rewarding, I'm sure readers are getting weary of the same format. During that time I've just been idly saving things to that list when I come across them.

Seventy-one items.

That's one a day for over two months.

Now, sometimes I'll pick one of those and, upon re-reading, ask myself what the hell I was thinking saving that garbage. Okay, well, no, if it's truly garbage then I'd have fun stuffing it in a trash bag and kicking it to the curb (that is, tearing the article apart in here). Point is, sometimes I'll get something that's not even worth ragging on and skipping it entirely, because, I don't know, maybe I was drunk or in a really different frame of mind when I saved it, or perhaps the world has moved on by the time it comes up.

But that happens rarely, and besides, I'm adding new links all the time, if irregularly. Writing to prompts is meant to help me catch up when the well's starting to run dry, but right now it's overflowing and eroding away the topsoil -- to stretch a metaphor beyond comprehensibility.

And so I used my RNG and it came up with an article that I saved fairly recently. Let's take a look, shall we?

Want To Rewire Your Brain For Meaningful Life Changes? Do These Things Immediately

You know, the whole "rewire your brain" thing has bugged me, at some level, for some time. Saying things like that, or "we're hard-wired to (do whatever)" is to use a particularly misleading metaphor.

Obviously, we don't have actual wires (nerves, sure, but in the brain, they're not the same thing as wires), and there have always been comparisons between the brain and some sort of technology. When clocks were all the rage, people used timekeeping metaphors. In the industrial revolution, it was machinery; some of this remains in our lexicon, like when someone likens the thinking process to "gears turning."

The brain has also been likened to a computer with its processors and hard drives (also a misleading metaphor), and I guarantee you if quantum computing takes off you'll have people talking about superpositions of brain waves and collapse of the mental wave function -- that is, when more people become familiar with the lingo of quantum physics.

But, whatever. The human brain is notorious for not being able to understand itself, so metaphor it is.

The simplest, most direct way to be smart is to build deep knowledge about things you care about.

This sent up red flags for me. I know subtitles are just there to catch our attention, and it worked in this case -- but I'm deeply aware that "knowledge" isn't the same thing as "intelligence." The other day we had the prompt about trivia, and that's a perfect example: being able to rattle off fact after memorized fact doesn't mean you can synthesize these facts into something greater, which to me is the core of intelligence. Of course I would say that, since my memory is shit but I like to think I'm smart anyway.

You are the architect of your brain.

Oh, now the metaphor shifts from electricity to buildings. Well, I suppose one can rewire a structure, too.

It turns out you can teach an old dog new tricks.

And now it's about dogs. Dude, pick a metaphor and stick with it.

Until recently, the conventional thinking was that our brains were hardwired at birth and therefore unchangeable.

But the good news is that our brains are constantly being reshaped by our daily experiences.

Back to the wires again. Whatever. I don't know how anyone ever thought things were "hardwired" when it was so obvious that experiences shape thoughts that this became the basis for psychotherapy.

Anyway, we're only four sentences into the article and I've already rambled on too long. I'd suggest actually reading the thing, not because it's particularly well-written (it's obviously not) but because the core message is one I believe in. I think it's best expressed in the article itself by this line:

It pays to crave and keep an open mind. Incredibly smart people aren’t always born that way, but rather are constantly working to improve their intelligence.

Though I think even that is a bit misleading, because you don't have to be a genius to be open to new ideas and experiences.

The article goes on to suggest that we consolidate what we learn through... writing.

Blogging is a great tool for reflection and sharing what you’ve learned, even if you don’t hope to make a living at it. And it’s free.

Writing expands our vocabulary, which has been shown to be directly correlated with success.

For various definitions of "success," I'm sure. Here we fall into the usual trap of pandering to outcome-focused learning. One sure way to piss me off, if I were a teacher, would be to ask me, "Why are we learning this? What use is it?" The future use is irrelevant. The important part is the learning itself. I'm not a teacher, though, because I'd probably want to kick the ass of whoever asks that, and that's frowned upon for some reason.

Point is, that's what I've been doing, or trying to do, here: finding things that are, or could be, interesting, and then writing about them.

I must be a genius!

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