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.

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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.

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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.

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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.

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October 12, 2024 at 8:39am

#1078145

A Few Observations

Tomorrow, I intend to get back to my usual annoying self here by posting a Revisited entry. As for today, well, I'm still in recovery, so I'll just summarize some of my observations on the trip that I may or may not have already blogged about (because I can't remember and can't be arsed to re-read what I've already posted).

I gained a greater appreciation of the intricacies of French wine. Contrary to French beliefs, it's not necessarily superior to other countries' wine, but it does take a different approach. Not that I know a whole lot of details; that takes years of study and practice. I don't mind practicing.

It's popular in some circles to tout the general superiority of Europe over the US. In other circles, the opposite is the case. As it turns out, in my opinion at least, that they're just different. Europe does some things better, like public transportation and healthcare delivery. The US does other things better, like... um... well, I can't think of anything offhand, but I'm pretty sure I would if I weren't so tired. I guess we invented the Internet and GPS, so there's that.

There are whole sections of Brussels where there are more chocolatiers on each block than all other businesses combined.

There are others that are pretty much entirely restaurants.

I gather that it is easier and takes less time to become a medical doctor anywhere than it does to become a chocolatier in Belgium.

Real estate prices (both rent and buy) are amazingly cheap there, compared to the US. At least in the country and small towns. And that's with a denser population.

Unlike my above comment about French wine, Belgian chocolate really is the best. Sorry, Switzerland. At least you won't send assassins after me for that observation. ...Right?

The old joke about how Brits think 100 miles is a long way, while Yanks think 100 years is a long time? It applies to mainland Europe as well. You just have to convert to SI units.

Best quote of the trip, by a Belgian tour guide, uttered between bites of a Belgian waffle topped with whipped cream and chocolate: "You Americans eat like you have free health care."

So that's all I'm going to post about the trip... unless, of course, something else reminds me of a thing I did there. Like I said, tomorrow, we're back to our regularly scheduled programming.

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