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 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.

1 Like

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.

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

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

August 9, 2022 at 12:03am

#1036259

Camus For You

A bit of philosophy, without much in the way of commentary from me. Okay, maybe a little.

The philosopher who resisted despair
Albert Camus and the search for solace in a cruel age.

I've been seeing quite a bit about Camus recently. I don't believe the universe is trying to tell me something, but perhaps it's time to listen anyway.

The point of the talk was to say that the entire Western world lived in a civilization that elevated abstractions over experience — that ultimately removed people from the reality of human suffering.

The "talk" in question was nearly 80 years ago, and it's possible that these things have only gotten worse since then.

For Camus, nihilism wasn’t so much about belief in nothing; it was about refusing to believe in the world as it is. And killing in service to some idea is just as nihilistic as believing that nothing is true and therefore everything is permitted.

It's hard to wrap my head around this idea, so I can't say if I agree or not.

The immediacy of a war or a natural disaster collapses the barriers between us because it’s so clear what has to be done. And while nothing redeems a tragedy, there’s at least some comfort in the solidarity that emerges from it.

Yeah, that doesn't happen anymore. It happened for about 15 minutes after word of 9/11 spread, but since then, nope. It's all about our differences now.

Camus always said that he was pessimistic about the human condition and optimistic about humankind. Maybe that’s a contradiction.

If so, that's one that I can understand. I simultaneously hold the belief that we're utterly boned, and that we're going to get it together enough to fix some things that really matter. Of course there will always be problems—and how boring a world it would be if that weren't the case?

Camus always returned to the myth of Sisyphus as the model of human defiance. The problem wasn’t that Sisyphus had to roll his boulder up a hill forever; it’s that he had to roll it alone.

And that's the metaphor for what I was just talking about.

The article is much longer and has some interesting points, though, again, I can't say I fully agree. I'll just leave you with one Camus quote that didn't make it into the article, but that always stuck with me:

In the midst of chaos, I found there was, within me, an invincible calm. In the midst of winter, I found there was, within me, an invincible summer. And that makes me happy. For it says that no matter how hard the world pushes against me, within me, there's something stronger – something better, pushing right back.

Perhaps I should pick up some of his books. Not ready to tackle the original French, though.

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