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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February 11, 2022 at 12:01am

#1026442

Drawing it Out

I can't draw.

Have you fallen for the myth of ‘I can’t draw’? Do it anyway – and reap the rewards

That is not completely true. I started out my career as a draftsman -- putting together technical drawings, plans to be used for construction. We transitioned later to CAD, but at first it was pencil, ink, compass, straightedge.

Those skills don't really translate to sketching or artwork, though. And my attempts at those more free-form styles have usually ended in abject failure. I have no sense of proportion, and almost nothing I've ever drawn has had any kind of aesthetic value.

Drawing is a powerful tool of communication. It helps build self-understanding and can boost mental health.

Insofar as becoming good at anything can provide a boost to mental health, I won't argue with that.

But our current focus on productivity, outcomes and “talent” has us thinking about it the wrong way. Too many believe the myth of “I can’t draw”, when in fact it’s a skill built through practice.

Pretty much everything is a skill built through practice. Writing, for example. But while I think I can write, that came to me with relative ease, whereas drawing is something that I've practiced and never been able to quite "get."

We're all good at different things, but the two things I wish I were better at are music and drawing.

I wanted to take art classes, but all of them required some basic knowledge before they'd teach you anything -- which is a like expecting me to know quantum physics before I take a class in quantum physics.

And those popular "learn to draw" tutorials were worse than useless to me. "It's as easy as 1-2-3!" For example:

Learn to draw: a duck

1) Draw an oval.
2) Draw a smaller oval.
3) Add bill, eyes, feet, and detailed feathers.

I could do 1 and 2 just fine. And then I'd give up in frustration because I had no idea how to do 3.

Devoting a little time to drawing each day may make you happier, more employable and sustainably productive.

I don't know about that. I'd doodle when I was bored in class, which was most of the time. One time I drew a face with horns in the corner of a notebook. The teacher came by, glanced at it, and took me aside. "Explain this," he demanded. It was only then that I realized that I'd quite accidentally drawn a passable likeness of him, only with the added horns.

That was the only time I've ever been able to draw a recognizable face, and, as I said, it was completely by accident. Perhaps my subconscious had different ideas, but consciously, I actually liked that teacher and hadn't intended to disrespect him.

Point being, if anything, it made me less happy.

While I'm at it, fuck productivity, especially sustainable productivity.

The article goes on to list a few supposed benefits of drawing, which is the kind of thing that usually annoys me. One should not need "reasons" to learn things. "When are we ever going to use this stuff?" is a question that means I could never be a teacher, because anyone asking it would be escorted out of the classroom.

I'd love to learn to draw. And I don't need a reason other than the satisfaction of creating something visually interesting.

Upon reading the article, I think it's likely that I do have a mental block. As I said, the one time I drew something recognizable was pretty much subconscious. I have my issues with what's written there, but there might be some good ideas. Perhaps it's not too late to give it another shot.

By the way, remember yesterday when I ranted about the sportsball game being scheduled for the day before Valentine's Day, and how that might affect grocery shopping? And how I said "it's a good thing they're not predicting snow here?"

Guess what's now in the forecast for Sunday morning. Go ahead. Guess. Hint: remove one of the spaces in the first sentence in this current paragraph.

Can I use this power for good instead of evil? Let's find out: "At least Halle Berry won't come to my door, wearing the catsuit, carrying two growlers full of fine craft beer."

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