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.

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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

May 1, 2021 at 12:03am

#1009449

Liberating the Angel

PROMPT May 1st

Write about one (or more) of your creative idols. Who do you look up to? Whose work are you most inspired by? Why?

Michelangelo once said, "I saw the angel in the marble and carved until I set him free."

Some people, like that famous Ninja Turtleartist, just have the talent. Almost anyone can learn the mechanics of art, but the essence of it is more... ineffable. Sometimes it seems like a superpower, or something supernatural is involved, but I believe in neither of those supers; I just think true artists have something that is missing or stunted in the rest of us.

So it is with this guy I know named Dave.

Dave is the brewmaster at a local brewpub. I can generally take or leave art; dance doesn't do anything for me; music can move me, but I can't say I fully understand it. What I do understand, though, is beer. It's art in liquid form; at its best, it's sublime and subtle, a whole that is far more than the sum of its parts.

As with other kinds of art, almost anyone can learn the mechanics of brewing. I've never bothered, myself; as I have no talent whatsoever in any other art form, it would just be a waste of my time to try when there's so much great beer out there made by people who do have talent.

But just as Michelangelo saw the angel when everyone else just saw a block of rock, Dave has the superpower of being able to take different basic ingredients of beer and just know what the final result will be like.

Which is not to say that everything he does deserves to adorn the Sistine Chapel, or whatever the beer equivalent of that is. He's made some... questionable... brews. I tried one just the other day, a beer brewed with hemp. Hemp is a botanical cousin of hops, so you'd think, well, maybe the two can be interchangeable? But you'd think wrong.

My first experience with hemp beer was at a brewpub I stopped by in West Virginia, not far from the Mothman statue

(which, incidentally, would probably make Michelangelo groan in his grave). As is my usual practice, I got a flight of samples of several of their beers, including the hemp one. Most of the beers were okay. The hemp beer tasted like the Mothman himself had pissed it out after a particularly busy night of scaring the shit out of the locals.

I looked around and saw all the hipsters gathered at the various tables (this was, of course, in the Before Time); they were pouring that foul swill into their beards and nodding like this was the Second Coming of Beer Jesus.

The one place in the world where I feel like I actually fit in is at brewpubs, but even then, sometimes that culture eludes me.

Anyway, so Dave made a hemp beer. After the Mothman pee incident, I was understandably reluctant to order an entire pint of the stuff; fortunately, you can usually get minuscule samples at brewpubs, just to check whether you like it or not. It's a Bad Idea to take advantage of this practice by requesting samples of everything on the menu; if you're going to do something like that, fucking shell out the few bucks for a flight. Don't be That Guy.

I sampled Dave's hemp beer and... well, it wasn't terrible. Which, given my previous experience with that particular style, is high praise. I'd never buy a pint of it, though, and since there weren't too many other people around, I couldn't tell if the local hipsters had decided that the idea of putting the plant from which cannabis springs into beer was the Best Idea Ever. I hope not. This is actual Virginia, where we don't usually marry our siblings.

Point being that even the few styles of beer that I generally avoid can't help but be touched by Dave's magic. One of these days I might even work up the courage to try his non-alcoholic beer. Maybe. Probably not. But I'm sure it's a fine example of that useless style.

As for the other questions, well, one of the many, many joys of getting old is that, as time goes on, there are fewer and fewer people to look up to. You get to watch your heroes die off, one by one; and the ones that are left, you realize are just people after all; people with talent, maybe, but otherwise just as flawed as the rest of us. So I don't idolize people. I respect their work, if appropriate, but I don't worship celebrities.

I'd have to say that, when it comes to writing, nothing has inspired me more than the work of Leonard Cohen, which, as I've noted before, I encountered first in poems before I found out he was also a songsmith. But even there, I soon found that my writing tone was going to have to be far less depressing (I mean, whose isn't? Apart from Morrissey) and that I'm more drawn to humor than to existential angst.

But hey, sometimes existential angst can be funny.

I mean, I still have this image in my head of the Mothman pissing out some sort of beer analogue that made it into the Point Pleasant brewpubs. And that's comedy.

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