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:

Turkey DrumStik

about 11 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

September 27, 2018 at 1:50am

#942077

Ad-verse Reaction

I hate ads.

Now, I don't mean that the way most people do. Most people will say they hate ads, and then continue to watch TV. They put up with the ads anyway. A necessary - or at least convenient - evil.

Other people deliberately watch the Big Trademarked Football Game in February just so they can see the really expensive ads when they come out. I can no more understand these humans than I can the ones who love camping in the cold. Mind you, I'm not ragging on them. To each their own and all that. I just don't understand.

So I've never had cable TV. What few shows I used to watch, long ago before the internet, I could pick up with antennae and grumble my way through the ads. Star Trek: The Next Generation, for example. As for cable, I never could wrap my head around paying an assload of money every month for something I would rarely use, and, despite paying for it, still have to put up with commercials.

But eventually even on-air commercials became too much for me to deal with, and I swore off TV entirely. Not that I didn't have a picture box; it's just that its sole purpose was to play VCRs and, later, DVDs. I filled my leisure time mostly with computer games - so don't think this is some screed against popular entertainment in general.

Streaming has brought me back to TV. There's Netflix, of course, with a reasonable monthly fee - and the only ads I get are for other videos, which is fine; it's like previews before a movie. And Amazon offers shows that you can pay for, also without commercials. I'm still paying less than I would for cable. If they have a show, they usually release the episode the day after it airs on commercial TV, which is fine with me.

But now, streaming services have been popping up all over, and usually, you can only get a show on one of them. This is getting to be a problem, as pretty soon it'll be too much like cable - too many choices, few of them worth it. So I have Netflix, and Amazon Prime (for which I get enough free shipping by itself to justify the annual fee), and CBS All Access.

That last one is problematic. I got it for one reason and one reason only: Star Trek: Discovery. But there's another show I like called Salvation, which is also offered on CBS. Good writing, science fiction, and they have an actual scientist consulting on the science (astronomer Phil Plait, whose work I've been following for years). Anyway, the problem is this: In the middle of the show - not at the beginning or end, as some other streaming services do - they interrupt with an ad for another show, even though I'm paying the higher "ad-free" price.

Even that wouldn't be so bad if the ad were for other science fiction shows. I mean, they know exactly what I watch on their site, and it's almost exclusively SF, so they could target their market. But no - the ads are for, if you've seen the movie Idiocracy, the equivalent of Ow! My Balls! - shows like Survivor and Big Brother.

Shows, in short, that I have never, and will never, purposely watch.

So - do I cancel my subscription, and miss out on Disco? Or put up with the Ow! My Balls! ads?

I feel like they'll keep doing it if they can get away with it, and I don't want to encourage that sort of thing. But I really like Star Trek...

Well, I'll tell you what - first time I see an ad for a home appliance, or cat food, or tampons, or anything like that, I'm out of there. Until then, I'll put up with it.

1 Like • Comment • View 3 comments

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