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').

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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

November 2, 2018 at 1:05am

#944696

In a Sense

"Did you know that butterflies taste with their feet?"

No, but I knew that butterflies taste horrible.

That's one of those things that you can only learn from experience. Oh, you can say it all you want, but unless you've actually zipped down the road on a bicycle, mouth wide open to gasp in the sweet spring air, only to suddenly have your cakehole suddenly filled with old shoes, industrial waste, aluminum, and ground fingernail, you probably think butterflies are magical flying flowers or something.

To be fair, there might be good-tasting butterflies out there. I know that some birds discriminate between the various species of fluttering insect, so, presumably, there are some that are delicious - at least to those birds. I have no desire to find out for myself; my taste buds exist for the primary purpose of discerning certain nuances in beer, wine and scotch, few of which sound particularly appetizing to the uninitiated. "This fine single-malt has deep flavors of oak, leather, peat, tobacco, and iodine." The challenge is to translate that into a phrase that might actually sound good to someone who's never had the transcendental experience of actually sipping whisky. Scotch aficionados, on the other hand, would be all over that.

Periodically, most writers have to be reminded to use other senses besides sight and hearing. One exercise from NaNo Prep comes to mind: "Describe a setting. Use all five senses." The result usually reads like an exercise in using all five senses, but it's good practice anyway.

Problem is, we don't have five senses. This is just one of the many lies they tell you in public school and, I presume, private school also. Well, not really a lie, actually - it's just that kids aren't exactly good at nuance and subtlety. One of them accidentally eats a butterfly, and it's all "Yuck! Hork! Patooey! Gross!" with no stopping to experience the full range of sensory experience on their tender palate.

The five "classic" senses - sight, hearing, smell, taste, touch - are really all just touch. Different specialized cells in the tongue and nose pick up on compounds released from the butterfly - or whatever - that brush against them. Hearing is the interpretation of rapid pressure variations in the air (or water); the sound waves tickle our eardrums, and a Rube Goldberg contraption of bone, muscle, and nerves interprets the result. Sight is similar, only with photons instead of sound waves. It's all, ultimately, touch.

Moreover, we have other senses than the classic five. A sense of balance, unrelated to the "touch" we learn about in school, is also unrelated to hearing - most deaf people can balance just fine - but relies on organs in or near the ears. There's the sense of proprioception, which is a fancy word for knowing where parts of your body are in relation to each other; you can raise your hand over your head, out of sight, and not lose track of it (we tend to take this one for granted, but it's a legitimate sense).

Some other animals have different senses, leading me to wonder whether a butterfly is actually tasting with its feet; perhaps it is smelling, or maybe it's using a sense we don't have a word for because we're not butterflies and our perception is thus limited. Sharks have at least three senses that we don't have, including the ability to sense electric fields, a feature shared with some other fish. Bats famously have hearing that's so sensitive that it might as well be sight for all practical purposes. Even common domestic cats (otherwise known as Our Overlords) have a sense that sort of combines taste and smell.

So yes, it's a good idea to feature the underutilized senses in writing. But if, like me, you're a fan of fantasy and/or science fiction, the real fun comes in making up different senses and attempting to translate them to our limited experience. How, for example, would you describe the sight of a species whose visual organs can detect different sections of the electromagnetic spectrum? Hearing that exceeds our own range of vibrational wavelengths? An eel following an electric current? The taste of a recently-discovered moon?

I know when I write, I tend to be movie-like - sights and sounds, with other senses mostly just available through dialogue or inference: "Ewww, what died in here?" Or a description of Our Hero's face twisted up after swallowing a shot of gin.

I should probably work on that.

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