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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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November 28, 2021 at 12:01am

#1022439

Life, The Universe, and Everything

If you can write about something, is it really intangible?

PROMPT November 28^th

Hopefully a little change of pace tonight. Write about something intangible: faith, magic, energy, power, or creativity.
Just choose one topic, and write about it.

Well, sure, I suppose it's still intangible if you can write about it. Consider a hole, for example. A hole is defined by its surroundings. You can't meaningfully point at a spot in mid-air and go, "That's a hole." It's a thing, but it's not a thing, at the same time. Thinking about it too hard makes your head hurt.

I guess, though, I'm limited to one of those five topics in the prompt, which is too bad, because I've been mulling over a rant about the abstract concept known as "freedom." I guess that one can wait.

So let's talk about energy.

As far as I can tell, energy has two distinct, separate definitions:

1) The capacity to do work;
2) The capacity to do work.

The first definition is physical. That is, it comes from physics. "Work," in this context, is force exerted over some distance. (Energy is also equivalent to mass, per Einstein's well-known formula). For example, the force required to move your car over a flat surface for, say, a mile, can be computed, and that force is (usually) supplied by energy produced by the car's engine, be it internal combustion or electric. Unless said engine isn't working, in which case it's supplied by your muscles and cuss words.

This kind of energy has several modes, such as: kinetic, potential, thermal, chemical, nuclear, etc. The important thing to know about energy is that it requires a gradient in order to perform the work. Solar energy works because the Earth's surface is cooler than that of the sun. Well, usually; may not apply in Phoenix in July.

Even though it can be measured and quantified, it's still intangible. A hydroelectric dam doesn't only have water, fish, and dead mobsters behind it; it also stores potential energy, though it's not like you can see or touch such energy. When the water passes through a turbine, it does work on the turbine, which converts (some of) that potential energy, by means of kinetic energy, into electrical energy.

The second definition is physical.

In this case, by "physical," I mean related to your body. Isn't English wonderful? And "work," in this case, is what one commonly gets paid to do (or receive some other form of compensation for), such as digging a ditch, writing an essay, or fixing the turbine in a hydroelectric plant.

At times, you may get the feeling of having low energy, and at these times, you just wanna lie in bed and not do any work. In reality, your body continues to chug along, using its (mostly) chemical energy (first definition) to do fun things like keep your organs functioning. When that process stops, it's called "death," and at times of low energy that can look pretty good -- fortunately, though, at those same times, we lack the energy to stop our energy.

Confusingly, you also don't want to play at such times -- "play" being the vernacular opposite of "work." This unfortunately does not have a corresponding definition in physics.

There's an old, really bad joke, which was featured (spoiler alert) in Ghostbusters:Afterlife: "Why should you never trust atoms? Because they make up everything."

Now, I've never let the facts get in the way of a good joke -- or a bad one (especially a bad one). But as the joke was told by a certified nerd, I winced. Not because the joke was so bad (it was) or because it is so old (it is), but because a nerd should know better.

Atoms don't make up everything.

The sun, for example, doesn't have many atoms in it. It's so hot that the electrons are stripped from the nuclei, and what you have is a bunch of free-floating protons/neutrons and electrons. And an "atom" is defined as nucleus plus electrons. Physicists call this phase of matter a "plasma." As the Sun contains the vast majority of the mass in the Solar System (according to Wiki, 99.86%

), atoms don't make up everything.

Also, dark matter isn't atoms (as far as we know), and it makes up something like 85% of the matter in the universe.

But okay, here on Earth, the vast majority of matter is composed of atoms. (If it seems like I'm using "matter" and "mass" interchangeably, it's because I am, because I can.) Still, going back to that equation I mentioned earlier, all of those (roughly) 1.3*10⁵⁰ atoms can, in theory, be converted to an unimaginably large amount of energy.

Doing so, however, would ruin your day. Hell, they'd probably see the explosion in the Andromeda Galaxy oh, two and a half million years or so from now.

The point being that atoms don't make up everything -- but energy does.

So much for being intangible, I guess.

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