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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Brandiwyn🎶

a day ago
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

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 25, 2021 at 12:02am

#1022273

Putting the Attitude Back in Gratitude

I have to admit I've been dreading this one all month.

PROMPT November 25^th

We all have something to be thankful about. In the United States, we celebrate Thanksgiving today. Tell us what you are most thankful for.

It's not that I'm not thankful. It's that I have a visceral aversion to talking about it.

Even right now, it's difficult. I've written and then deleted several sentences already, because none of them seemed quite right to me.

And the temptation is definitely there to turn the prompt to something humorous. I'm going to go ahead and give in to that urge by linking to a special Thanksgiving poem I wrote a few years ago:

Alcohol, Hookers, and Blow

(18+)
A Thanksgiving thanks giving
#2175291 by Robert Waltz

Well. Now that that's out of the way, what am I actually most thankful for? Despite the poem's title and content, I've never been with a hooker or done blow (that's cocaine, for anyone not used to US slang). Alcohol, yes, sure, but is it really what I'm most thankful for?

I'm in a pretty good place in life. Not without problems, of course; everyone has problems. But overall, things are pretty good for me right now. There's a downside to that, though, which is that it gives me no incentive to try to improve my situation further, because any move I make to improve could backfire and make things worse. So while I'm thankful for all the beneficial things in my life, that wouldn't be the top of the list.

I suppose when it comes right down to it, the thing I'm most thankful for is existence itself. That may be a kind of cop-out -- after all, if I didn't exist, I wouldn't be here to experience gratitude. But I think that, by itself, is enough reason to be thankful. And it's not just my existence I'm talking about, either. No one can exist without reference to what's around them: air, water, sunlight, people, whatever.

I'm not doing anything for Thanksgiving this year. Well, not exactly. It's traditional to eat turkey, so I bought a frozen turkey pie to eat later. And nothing goes with turkey as perfectly as Beaujolais Nouveau, a delicious red wine that gets shipped here every year from France just in time for Thanksgiving (so thanks for that, France). So yeah, I'm going to be a cliché single person, eat the pie, and drink the bottle of wine. Probably while watching TV shows. Some might consider that pathetic. I do not. It is far less stressful than having to deal with other people.

After that, I'll go out to see a movie, because I can.

Existence? Maybe I spoke too soon. I think what I should be most thankful for is not having to abide by others' schedules and preferences. I get to do exactly what I want to today, not what other people think I should do.

And for that, I am truly grateful.

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