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 11 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 13 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').

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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 16, 2022 at 12:05am

#1032463

A Total Eclipse of the Moon

It is a night of a lunar eclipse. The actual full moonination (I just made that up) occurs at 12:14 am (yes, I keep track of these things), which would be when the lunar orb is fully within the shadow of the Earth. That puts it way high up in the sky, to use a technical phrase from astronomy -- perfect for viewing. Usually, during a lunar eclipse, the moon takes on a deep red glow, the reflected light of every sunset and sunrise on our planet simultaneously. It's all really very cosmic.

It's not all that rare; a lunar eclipse happens every few years. I'd say "once in a blue moon," but that would only make the confusion surrounding the definition of a blue moon that much worse. I've ranted about that before. I'll rant about it again. But not today. Common or not, I like to look at these things, especially now since it's the first one since I got new eyeballs.

Naturally, the sky where I am is completely covered by a thick blanket of clouds.

I should probably be more frustrated by this than I actually am, but -- though I hate it when someone goes "well, it could be worse" -- it could be worse. I got the news yesterday (Sunday) that my last remaining aunt, who had a stroke back in February, died over the weekend. She was in her 90s, so it's not like it was totally unexpected or anything, but still sad.

My cousin told me the funeral would be on Wednesday. I went online and started looking for near-last-minute transportation, because I still don't have a car. "Oh, you have to travel in two days? You must be desperate. Here, let us give you the highest price we possibly can." Usually I take a train to NYC, because it's a lot less stressful even if it takes longer. But the trains were booked solid. Still, one cool thing about our little airport here in Charlottesville is that they have daily nonstop flights to La Guardia. That's flights, plural. No puddle-jumping to IAD and getting drunk in the airport bar while waiting for your connection, and lots of options to choose from -- all remarkably pricey on short notice.

Right after I booked the ticket, he called to inform me that, well, no, the funeral would be held in two weeks instead. On Memorial Day weekend.

To his credit, he offered to reimburse me for canceling my really very goddamned expensive booked flight. I mean, seriously, I've flown to Vegas for less. Shit, I've flown to bloody England for less, but that was over a decade ago so maybe it doesn't count. It's pretty close to what I paid for a trip to goddamned Maui.

Problem is -- setting aside for the moment the absolute joy that traveling on Memorial Day weekend is -- I just can't go that weekend. My housemate has her own family plans, and has had them for some time. We have pets. Finding a holiday weekend cat-sitter on short notice is harder than watching an eclipse through the clouds, and I'm not leaving the cats alone.

So it looks like I'll miss the funeral. Hopefully I can take the trip sometime in June or July, go visit my cousin, see the grave where she'll be buried with my mom, her other sister, her brother, and her parents.

Still mostly cloudy here, though I just looked up and saw a single star. It looks like Mars but it's in the wrong place, almost directly overhead. I don't think it's Antares, either; that should be closer to the moon. I can't really tell what it is, because I don't see any other stars to get my bearings.

The whole neighborhood is dark. I think everyone's trying to look at the sky.

Ah. The clouds parted, briefly, and I got a glimpse of the blood-red moon.

Mission accomplished.

Oh, incidentally, the airline completely refunded my ticket price. If I were a total asshole, I'd tell him it was nonrefundable and collect the money, but I'm only a partial asshole.

I'm not entirely in the shadow, you see. Or, well, technically, right now I am, but at least I'm looking at the stars.

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