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

October 11, 2024 at 7:33am

#1078100

Beersday

As expected, my sleep schedule is all screwed up now, and it might take me a few days to get back to my normally-screwed-up Eastern Time schedule.

Fortunately, I don't have a lot of requirements for being at certain places at certain times, so I'll take as long as I need to recover. I do have a role-playing game scheduled for tomorrow evening, as is the case for most Saturdays when I'm not traveling, but that's about it.

One of my cats remembered me right away. The other remembered me after I gave her some food. "Oh, yeah, you're that human."

The true miracle of the trip, though:

When I did the Bosteels tour, as part of it, they gave me a Tripel Karmeliet boxed set—a bottle of that delicious beer, packed with two glasses (remember, most Belgian beers have dedicated glassware). There's nothing extra-special about the TK glasses; they're fairly normal tulip chalices, but with the beer logo printed on them.

The problem is, they're somewhat fragile.

My suitcases were already stuffed, so I spent the rest of the trip wondering how I was gong to get this boxed set home without everything breaking. Maybe I could mail back some clothes, so I'd have room? Mailing the beer to myself was right out, because of stupid laws. US Customs allows up to one liter of alcoholic beverage in checked luggage, and this bottle is under that limit, though of course you can't bring that quantity of any liquid in a carrion bag.

Well, I spent some time over there creatively rearranging stuff in my bags, and trying to ensure that the box containing the fragile, precious cargo was well-padded with dirty clothes on all sides (look, I wasn't about to waste time in Belgium looking for a laundromat). Still, I've done similar things before, a bottle of beer from England several years ago, and it came back all shattered. The beer soaked into the clothes, but I'm used to that and I was going to have to wash them upon returning, anyway.

So I wasn't holding out much hope. Between the rough handling of the bags, and the pressure and temperature differentials at flight altitude, I mentally prepared myself for returning home with a box of shattered glass and beer-soaked clothing.

But, somehow, the beer, and even the glasses, survived intact.

I should make a holiday of this, like how Hanukkah celebrates the miracle of the lamp oil lasting longer than it should have. October 10 will be Beersday for me from now on. It's even easy to remember, since it's 10/10 in both US and Euro formats, and it's also the birthday of one of my best friends.

Really, it's not even that big a deal, except that this was my only physical souvenir of the trip (like I said, suitcases were already stuffed, so no room for plastic Eiffel Towers or Manneken Pisses). Tripel Karmeliet is generally available in my area, albeit at exorbitant import pricing, and the glassware can be ordered online. I'm just glad my luck held out.

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