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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Turkey DrumStik

about 7 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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').

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

November 26, 2018 at 12:47am

#946312

Magic Want

In my life as an adult, I've never done anything I didn't want to do.

Now, before you call bullshit on that, let me explain. It's a matter of framing.

Let's talk about a dirty toilet, for example. As I suspect is the case with most humans, you don't enjoy cleaning the toilet. But let's further say that you can't afford to pay a servant to do it for you, and you're unable to trick your kids into doing it, but there it is, a toilet, in your house, and it's got... stains... in the bowl, and maybe on the seat, too.

You want a clean toilet, but there's no one else who can make it clean; consequently, it stands to reason that you want to clean the toilet.

I mean, if you didn't want to clean the toilet, you wouldn't, and you'd decide that you can live with a dirty toilet, after all.

You can apply this to anything in life, at least anything that's a conscious decision. (Though there could be an argument about exactly what constitutes "consciousness," I'm not wading into that quagmire in this entry.) A college professor assigns you homework; you're not thrilled about the idea of doing homework, but you would like to graduate. The dog begs for a walk; it's 20 below out there and windy as hell, but you wouldn't like the dog to pee in the house. You run out of liquor, and you don't particularly feel like going to buy some more; but you want liquor, so you do so.

Maybe when you're a kid, things are different. Sometimes you do what your parents say, even if you don't want to. Or maybe there are things, as a kid or an adult, that get forced upon you whether you like it or not, but in that case, you're not consciously doing something you don't want to do; you've been thrust into a situation you didn't want to be in, which is different.

Now, you can, if you want, choose not to see it that way. You can frame it as "I don't want to clean the toilet." But that's how you end up resenting things in life, and that's no way to go around.

So I've never done anything I didn't want to do.

My problem is actually the polar opposite of this: I very often don't do the things I do want to do, like decluttering the kitchen or going to the gym. I know I want to do these things, and others, and yet, somehow, I don't get around to them. And that's a problem I haven't figured out how to reframe, yet.

Or maybe I just don't want to.

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