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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 6 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.

1 Like

Adherennium Plotting something

about 8 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.

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.

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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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

May 20, 2021 at 12:05am

#1010459

Unalienable

PROMPT May 20th

What does happiness mean to you? Reflect back on a specific moment when you felt unconditionally happy. What circumstances made you feel that way?

So, yeah, I've ranted about happiness as a concept in here before, so some of this might be a rehash.

Perhaps because of the infamous meme posted on the internet by noted rapper DJ TJ, "the pursuit of happiness" has become enshrined in the actual (okay, the figurative) DNA of the US, along with discrimination and the benefits of working yourself to death.

A couple of notes on that, though:

1) Word meanings change over time. I mentioned here not too long ago that the meaning of "nice," for example, once meant something much closer to the sarcastic version today, as in "well, that's a nice pickle you've gotten yourself into." This is especially true for abstractions such as "liberty," or, indeed, "happiness." We -- well, I, anyway -- tend to think of it as people walking around with big smiles on their faces (assuming they're unmasked), probably without a care in the world, likely not very bright. But in its 18th-century meaning, the connotation was closer to prosperity, thriving, wellbeing

2) The same happy people walking around maskless and smiling without a care in the world tend to forget about the "pursuit of" part. The DoI never claimed happiness as a right, only the pursuit thereof.

Consequently, "the pursuit of happiness" can be more precisely interpreted as the right to self-determination, to follow one's own chosen path in life to the extent that a person is mentally and physically able, and not be, say, limited by one's parent's profession or, to extend it into more modern terms, being shackled by traditional gender roles.

By the way, I don't want to hear about Jefferson's apparent hypocrisy in penning those words while owning actual slaves with no self-determination. I live in Charlottesville, so I'm fully aware; and we've all seen Hamilton. Or if you haven't, there's a video of it on Disney+; go watch it already.

So. Happiness. Modern connotation. I think it's only meaningful in contrast to its opposite, kind of like how it feels so good when you stop banging your head against a brick wall. Thus it is inherently a fleeting state. Some people like to pretend they're happy in order to fit in. Others pretty much have to pretend they're happy, even if they're working a shit retail job and having to deal with ignorant assholes. Quitting such a job, provided one has an alternative means of support, is, incidentally, practically guaranteed to produce feelings of true happiness -- for a few hours, anyway.

As for discussing happiness on a personal level, I've already mentioned in here - several times - the sensation I call beerenity Last time, I think I described it as utter calm, but utter calm makes me happy, so it fits here as well. And I'm pretty sure I talked about the time I scored some Pappy van Winkle bourbon. Man, that was a good night.

But since I've already told those stories, and because I don't want to describe things of a particularly adult nature, things that used to occasionally induce happiness, in here, how about the time I quit my job (even though it wasn't a shit retail job) and started my own company?

Sure, I knew co-running a company would be a massive source of a different kind of stress, but in that moment, I felt freedom because my destiny wasn't tied to a boss. It's like I said above with self-determination.

That only lasted a couple of hours, though. Then I had to fill out forms and meet with an accountant, both of which inevitably give me a headache. And headaches are anathema to happiness.

In any case, I don't see happiness -- modern connotation -- as a goal. No, it can be the result of achieving a goal, but in my experience, chasing happiness for its own sake is counterproductive. It only comes as a result of something else. And even then, trying to hold onto it beyond its natural life often leads to despair.

But without the despair, how do you know you were actually happy?

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