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 3 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!

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

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').

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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

January 19, 2019 at 12:48am

#949965

Unnatural

Find a piece of nature that you can hold in your hand (leaf, twig, rock, berry, etc). Describe it as closely and carefully as you can. Use any means available to you to examine the object (magnifying glass, scale, all of your senses) and practice your descriptive writing skills.

"That's the problem with nature. Something's always stinging you or oozing mucus on you. Let's go watch TV."
-Calvin

The cycles turn, as they are wont to do, and so we have the four seasons: Cold, Pollen, Mosquitoes, and Dead Leaves. If you're lazy, Dead Leaves season extends well into Cold season, and so you can pick a crumply brown thing up off your deck without having to venture too far into the dreaded Outdoors.

I don't know how many microorganisms tracked into the house on this dead leaf, and I'm not about to get a microscope to find out. Probably some insects, too. Definitely some things that, if they were big enough to see, would bite or sting me.

At some point, probably during Mosquito season, this sample of detritus was, presumably, green. And then, during Dead Leaves, it might have become an interesting color, something on the low-frequency end of the spectrum like red, orange, or yellow.

Color, though, is a function of selective reflection of photons. It's not truly an innate property of an object. We call something "green" if, upon exposure to the right kind of light and in the presence of our optical receptors, it absorbs most of the wavelengths that we don't call "green." When it's brown, that means that a bunch of wavelengths are being absorbed, and we're seeing the others. If there's no light, there's no color.

This particular leaf is in the process of decaying into an amorphous chemical sludge. Certainly it smells that way. I'm probably helping it along by touching it. It's the fate of everything that lives, of course: its components to be recycled into something that's alive, only for it to die and allow the components to be recycled again.

Doesn't seem to have a point to it, does it?

Maybe that's the true beauty that eludes me in nature - that it really doesn't have a point. It's rather invigorating to believe that this is true.

Still doesn't mean I want to go out in it.

Thing is, though, we speak of nature as if it were something separate from us, when the simple truth is, we're just as natural as this leaf. Thus, anything we create - walls, hair dye, pyramids, rockets, computers, whatever - is also natural. The only thing artificial is the distinction we make between "human-made" and "natural." Where I spent my childhood, there were beavers, and they built dams. We, too, build dams. Termites create vast cities; so do we. Birds make nests, rabbits dig holes... these things are considered natural, and so too should be the things we create.

Possibly the most dangerous result of this artificial distinction is the idea that what's natural is good, while what's artificial is bad. Yeah... tell that to Socrates. Or anyone with sunburn-caused melanoma.

That's why I heap scorn upon products proclaiming that they contain "all-natural ingredients." Like, what, other products contain supernatural ingredients?

Anyway, time to let the leaf go, hopefully along with its slimy, gross microorganisms. There'll be more in a few months.

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