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!

a day 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

a day 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.

July 8, 2019 at 12:14am

#962268

Truth, Justice, Etc., Etc.

What fictional character do you admire? What specific characteristics do they possess that make them attractive?

I'm tired and I need to get up before dawn to take a friend to the airport, so I'mma go with the low-hanging fruit here.

Wow, I can actually feel y'all rolling your eyes. But I'm going to press on anyway.

Kal-El, as we all know, was adopted by an all-American couple who wanted kids but couldn't make 'em. That right there was enough to start the identification going.

Now, I should point out that identification isn't a big deal for me. I don't put myself in the protagonist's shoes. Doesn't matter to me what race, religion, nationality, sexuality, etc. that a character is; the only thing that matters is that it's a good story. (Yes, many Superman stories sucked, but remember, I was a kid when I started reading comics, and what's "good" has changed since then.)

He was also a farm boy. I was a farm boy.

Only child? Check.

Had to pretend to be something he wasn't? Check. Though, later, I'd start to wonder which identity was fake and which was real, but that's a more philosophical, adult conversation.

I mean, really, the whole superpowers thing was secondary. Sure - flying, super-speed, x-ray vision, etc. was cool. Costume was a bit poncy, but whatever; it was distinctive. What mattered, as I gradually became aware, was that regardless of superpowers, Superman always tried to be the best he could be.

During the "dark and gritty" phase of comics, Superman got a lot of shit - though not as much as Aquaman. Big Blue Boy Scout. Whatever. Sometimes idealism is a good thing. As much as I also like antiheroes, sometimes we do need an ideal. Of course, it's easy to do the right thing if you're a fictional character, and the writer can set it up so that whatever you do turns out to be the right thing. But ideals aren't meant to be realistic; that's why we make the distinction.

I don't think too many of us could honestly say that if we found ourselves as a god on an alien world, we'd always strive for that ideal. The temptation would always be there to use your powers for, if not evil, at least to impose your own version of "good" on the alien world. To stand apart, aloof, and utterly defeat whatever you see as "evil."

That he's not depicted as doing that is, I think, what makes the character a hero. Anything imposed from above is oppressive; we all have to find our own way. But we can all use a little help now and then.

I know my thoughts here aren't exactly new. Whole volumes have been written on the subject. Regardless, I've always liked the character, even when it wasn't cool to do so.

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