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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Write_Mikey_Write!

about 2 hours 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.

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Adherennium Plotting something

about 4 hours ago
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.

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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.

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.

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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.

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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.

December 8, 2021 at 12:03am

#1022945

Transcendental

Maybe you thought you could stop thinking about "gratitude" because Thanksgiving is over.

Sunday Firesides: Graduate From the Kindergarten Class of Gratitude

I'm no stranger to knocking out strings of words on a given topic in a few minutes. Hell, I do it every night. This reads like that. Only, presumably, these authors are getting paid, and I'm not.

Is there anything more innocuous than gratitude?

Puppies, maybe?

It’s one of the few values endorsed by solemn religious leaders and vapid lifestyle gurus alike.

I'm going to start using the phrase "weaponized adjectives" to describe this sort of blatant editorializing.

Unlike other virtues, which require going against instinct (e.g., you feel afraid, but decide to act with courage anyway), with gratitude, you simply lean into it; something good happens, you feel good, and you need only recognize that warm and fuzzy sensation.

No, if something good happens, I wait for the other shoe to drop.

Yet there’s more to gratitude than commonly countenanced — higher and harder shades of it to reach for beyond its elementary-level start.

“Advanced” gratitude retains delight in someone’s admirable qualities and small acts of service . . . even after the novel and noticeable becomes the ordinary and expected.

If you say so.

If elementary gratitude is instinctual; advanced gratitude is effortful. Whereas one is merely felt, the other is expressed.

Some people say that we only appreciate the things that we have to work for. I happen to disagree. I appreciate more the things that are handed to me, because they're so hard to replace.

If, in the kindergarten class of gratitude, the warm fuzzies of thankfulness are for the heart-lightening benefit of the individual alone, in post-graduate gratitude, they are used as a spur towards being better, doing better, giving back.

And that's fine, but it doesn't go far enough.

The next level, as far as I'm concerned at least, is being grateful not just for the good things that happen, but for the shitty things.

I'll call this "transcendent" gratitude, so I can weaponize that adjective, and because that's the only thing I can think of beyond "advanced" in the limited time I have to do this entry.

Say a tree falls on your house. That sucks, right? Especially if you're in it at the time. But if you survive, transcendent gratitude is being thankful for the opportunity to step outside your comfort zone, learn new things, figure out how to make an insurance claim, navigate the emergency room for the concussion it gave you, and how to get someone to fix it as quickly as possible -- all valuable life experiences.

Or you loan a friend $5000. She doesn't pay it back. You should be grateful for that life lesson: don't lend money to friends -- or pick better friends -- or never trust her again. Or, in my case, all three.

Or your car breaks down on US 50 in Nevada, with no cell service, 110F+ temperatures, and no help within 200 miles. Be grateful for the opportunity to practice all the skills you've prepared for just such an occasion.

Now. I'm not saying I practice transcendental gratitude, at least not on a regular basis. Frankly, I'd rather give up and bitch about it. But bad shit happens, inevitably, and you can either bitch about it and give up, or be grateful for the learning experience.

And see? I can knock out a monologue about gratitude in less than a half hour, too.

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