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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March 6, 2022 at 12:01am

#1028405

Dedicated Indoorsman

"That's the problem with nature. Something's always stinging you or oozing mucus on you. Let's go watch TV."
-Calvin

This Is No Way to Be Human
We now occupy a nearly natureless world.

I could quibble about the definition and connotation of "nature." We are absolutely a part of nature, and thus the things we construct are also a part of the natural world. Is a beaver dam unnatural? A termite mound? A bird's nest? Just because it's built by an ape doesn't mean it's not part of nature. And some things that we humans create can be as beautiful or terrifying as a wilderness untouched by hands with opposable thumbs.

But, okay, fine, I'll take for the sake of argument the stance that anything created by humans is "artificial" and anything else, including artifacts created by other animals, is "natural."

As an aside, though, let's try to remember that "natural" isn't always "beneficial." My cigars contain all-natural ingredients. Poison ivy is natural. Most viruses are natural. And so on.

Many of us invest hours each day staring at the screens of our televisions and computers and smartphones. Seldom do we go outside on a clear night, away from the lights of the city, and gaze at the dark starry sky, or take walks in the woods unaccompanied by our digital devices.

I like computers and smartphones. TV, not so much, because of the interminable stupid ads. Those get my ire up way more than being in a fluorescent-lit cubicle ever did.

Most of the minutes and hours of each day we spend in temperature-controlled structures of wood, concrete, and steel.

Goddamn right I do. Okay, so I do like to sit on my deck with the trees around. Yesterday I saw a whole flock of deer cross the yard. But in the past, I've been perfectly content to sit inside, and also always when it's cold. It's cold a lot. Give me temperature control. We developed technology for a reason, and that reason is to be fucking comfortable.

We have created a natureless world.

That's a matter of opinion and perspective. Like I said above, my painted drywall is no less "natural" than a tree, as it was created and installed by natural creatures.

It was not always this way. For more than 99 percent of our history as humans, we lived close to nature. We lived in the open.

Yes, and we had the brains to stop doing that shit. Not all early humans were "cavepeople," but what's the real difference between a cave and a house, except most houses at least have windows you can look out of?

Over the large majority of our 2-million-year evolutionary history, Darwinian forces molded our brains to find kinship with nature, what the biologist E. O. Wilson called “biophilia.”

I think that's misleading. "Darwinian forces" (COME ON) molded our brains to be able to make tools, then tools to make other tools, then tools to construct all the wonderful and terrible things we've built over the millennia.

In other words, if [deity] had not intended humans to fly, [pronoun] wouldn't have given us the ability to build airplanes.

But okay, fine, perhaps we sometimes go too far with it.

In 2004, the social psychologists Stephan Mayer and Cindy McPherson Frantz, at Oberlin College, developed something called the “connectedness to nature scale” (CNS), a set of statements that could be used to determine a person’s degree of affinity for nature... Some of the statements of the CNS test are:

And you're damn right I'm going to comment on said statements.

I often feel a sense of oneness with the natural world around me.

I feel a sense of oneness with my laptop and smartphone.

I think of the natural world as a community to which I belong.

I think of the natural world as including the things we've built.

When I think of my life, I imagine myself to be part of a larger cyclical process of living.

Well, okay, I can't argue with that one.

I feel as though I belong to the Earth as equally as it belongs to me.

Oh... if only the Earth belonged to me. There'd be some changes, lemme tellya.

I feel that all inhabitants of Earth, human and nonhuman, share a common “life force.”

There is a light side, and a dark side, and it binds the universe together. I mean, again, COME ON.

In the more frenzied and tech-heavy times of today, we require more effort to creep out of our close and crowded houses. But the poet Mary Oliver succeeded. In her 1972 poem “Sleeping in the Forest,” Oliver writes that she “slept as never before, a stone / on the riverbed, nothing between me and the white fire of stars / but my thoughts, and they floated / light as moths among the branches / of the perfect trees … By morning / I had vanished at least a dozen times / into something better.”

Yeah, I've slept outside, too. Or, rather, tried to. It was cold. The ground was hard. Shit crawled on me. The next morning, I had a massive headache.

Don't get me wrong: I appreciate what we're calling "nature" here. I just prefer to appreciate it from a nice warm room.

My most intense experience with nature occurred a number of years ago on a small island in Maine. A family of ospreys lived near our house on the island...

That's nice. Usually nature's birds just naturally deposit their shit upon me.

But I do think that we need to be more mindful of what this technology has cost us and the vital importance of direct experiences with nature. And by “cost,” I mean what Henry David Thoreau meant in Walden: “The cost of a thing is the amount of what I will call life which is required to be exchanged for it, immediately or in the long run.”

Anytime someone tries to convince me of something by quoting Thoreau, I become convinced of the opposite.

In summary, it's as if we've built ourselves a splendid mansion, and now you want us to go live in the mud. No, thanks. Oh, sure, in small doses, I agree: "nature" can be beautiful. But I think a lot of the problems supposedly caused by "artificial" surroundings are due to other concerns: when you're in that fluorescent-lit office, for example, you're generally focused on work, which can be stressful, but when you're walking in the forest (unless you're a lumberjack), you're not thinking about work.

Other people have other priorities, of course. But as for me, out here sitting on my deck because it's the first midnight in a long while that hasn't been freezing cold (which I define as anything under about 55F), I'm perfectly happy to have my electric patio heater plugged in and keeping me warm while still technically being "outdoors."

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