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 9 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 11 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.

June 29, 2021 at 12:40am

#1012703

The Forbidden Door

Entry #8 of 8 for "Journalistic Intentions"

[18+]. I figured I'd better get in this final entry while I have time.

Doorways I Haven't Opened

There are several oft-repeated proverbs that irk the hell out of me. For example: "There is nothing new under the sun." I don't care that it comes from the Bible; it's not only too cynical even for me, but demonstrably untrue. Oh, sure, it might have been true 2-3000 years ago when things were relatively stagnant, but... well... I've been traveling on back roads in a car, following directions from Google Maps on my Android, and staying in basic but comfortable hotel rooms. None of these things existed 200 years ago, and hell, Maps and smartphones weren't around 20 years ago. They are new, and really, "under the sun" may be poetic, but it doesn't exactly mean much (at night, are we "over" the sun?)

Of course, one could argue that it's a restatement of matter-energy equivalence, of how neither can be created nor destroyed, only transformed. I don't give the writers of Ecclesiastes that much credit, though. And new stuff happens all the time, both good and bad. Atomic bombs, airplanes, robots, comedy routines, Coke, electricity transmission, trips to the Moon, and so on.

Another proverb I can't stand is: "That which does not kill you makes you stronger." Pretty sure that one comes from Neitzche, lest you think I'm only ragging on Bible quotes here. Also demonstrably untrue, and it irks the living hell out of me; I've spoken before of my WW2 veteran uncle, who survived the war only to suffer PTSD for nearly half a century afterward, until his death. Anyone who thinks he's stronger for that gets my deepest scorn.

Neitzche was probably referring to his epic mustache. Pretty sure it had a life of its own, and he had to be stronger just to support the weight of it.

But of all the stupid duck-billed platitudes bandied around, one of those that I most utterly despise is: "When one door closes, another opens." I mean, what even? Better hope you're not on an airplane when that happens. This one is attributed to inventor Alexander Graham Bell, who might be forgiven for it because he lived before airplanes were invented. In any case, I don't care who made it up; this is nothing more than wishful thinking. I mean, other than the airplane thing, consider: prison cells and hotel rooms. I hope when I close the hotel room door that another one doesn't open.

But even if you insist on keeping it metaphorical, death can be considered the closing of a door, and that's pretty final. Unless of course you indulge in wishful thinking.

No, sometimes another door does not open, nor does a window; and moreover, sometimes you don't really want it to.

Still, that's probably tangential to the prompt, but far be it from me to take things too literally, right?

I have a strong urge to go to forbidden places. Not, like, Chernobyl or anything like that (though that might be cool), but, like, penthouses, steeples, the tops of suspension bridges, the torch of the Statue of Liberty, the inside of Elizabeth Tower in London -- that sort of thing. Yes, mostly high places. I have no fear of heights; in fact, I love the views. But it would only be made more awesome if it's a place not generally open to the public.

I have heard that Elizabeth Tower, at least, can be visited -- if you're a UK subject. From what I understand, you just have to arrange it with your MP. I've seen pictures of the inside, including the world-famous Big Ben (just to be clear, Big Ben is the bell; the tower itself was only named in 2012). Point is, though, Americans are shit out of luck there. And the S of L torch has been closed off for, I don't know, I can't be arsed to look it up, but something like a century.

And most of those other places are just plain inaccessible. Daredevils have done some of them, mostly getting caught and punished for their crime of wanting to go someplace They don't want you to go. These attractive, forbidden places are all behind locked doors. Doorways I haven't opened, and probably never will.

Trip update: Decorah, Iowa. Remember yesterday how I was talking about places that are named for things that no longer exist? Decorah, it turns out, was named after a Native American leader. You can guess what happened to their tribe. Hint: they aren't in Decorah.

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