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 6 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 8 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.

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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.

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.

January 20, 2022 at 12:03am

#1025049

Freeze Peach

A provocative one from "JAFBG"

[XGC].

Should there be limitations on free speech. And if so, what should those limitations be?

Of course there should be: anything I don't like should be banned.

Okay, no, of course I'm joking. Still, it seems some people actually think that way. This is not a jab at any one political side; it's an actual rare case of "both sides are bad," or, rather, extremists on both sides are bad. True freedom begins when you realize that it's better to hear opinions you don't like, and to have the opportunity to refute them, than it is to ban them.

Keeping in mind that I am and have always been a US citizen, and this is therefore written from a US point of view.

There are of course some limitations on free speech. We've seen what happens when there aren't: internet boards getting filled up with ads, for example.

The limit most often trotted out involves shouting "Fire!" in a crowded theater, on the theory that doing so could lead to injury or loss of life in the ensuing panic. (The actual court decision adds the word "falsely" to that -- just so you know, in case you're ever in a theater and you actually see something burning and want to warn people.) Never established, as far as I know, is if it's okay to shout "Movie!" in a crowded firehouse.

But the basic idea -- that some forms of unfettered speech can cause imminent harm, and should thus be proscribed -- seems to have some validity. It's the "imminent" part that leads to gray areas: there's a difference between "Trespassers should be shot" and "Let's shoot those trespassers now."

And there are always gray areas. That's why we have arguments.

As I see it, anyway, imminent harm isn't the only criteria. We've decided that public pornography isn't acceptable, for example. Or naming your business something containing a curse word (though I find it hilarious that there are actual Vietnamese restaurants named Pho King, because I'm secretly 12 years old).

The point is, yes, there are always limits, because speech isn't the only right worth protecting. Sometimes it comes into conflict with our other rights: life, liberty, privacy, whatever; and a balance has to be maintained there. Where exactly that balance lies is why we have courts and shit. And the freedom to argue about it.

Most importantly, private companies and other entities get to set their own limits on free speech, which I think a lot of people miss in their whining about being silenced. Someone comes into my house going "Hitler was right," and while I would never say that they shouldn't be allowed to say that, they'd be unwelcome in my house forevermore. In fact, I want to know if someone's a racist, classist, Judeophobe, sexist, etc., so I can know who to ignore, and I can't know that unless we give them free rein to express their idiotic opinions.

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