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 3 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 5 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').

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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.

April 30, 2021 at 12:02am

#1009392

Conscious Decisions

Well, this will probably be the last random article for a while; let's see what we end up with.

The Forest Spirits of Today Are Computers
We’ve made an artificially panpsychic world, where technology and nature are one.

Oh joy, panpsychism again.

Well... sort of.

Years before smart homes became a thing, I replaced all the switches in our house with computerized switches. At first, it was just a way to add wall switches without pulling new wire. Over time, I got more ambitious. The system runs a timer routine when it detects no one is home, turns on the basement light when you open the door, and lights up rooms in succession on well-worn paths such as bedroom to kitchen. Other members of the family are less enthusiastic. A light might fail to turn on or might go out for lack of motion, or maybe for lack of any discernible reason. The house seems to have a mind of its own.

I can see the appeal, but the tech just isn't quite up to Star Trek standards yet.

Also, to me, "seems to have a mind of its own" translates to "does stuff at random times because of power fluctuations, cosmic ray events, or loose wires." Never ascribe to consciousness what can better be explained by glitches -- unless, of course, you can get laughs by doing so.

Modern hardware and software have gotten so complicated that they resemble the organic: messy, unpredictable, inscrutable.

On a superficial level, maybe. I'd prefer to say that it's gotten so complicated that there's exponentially more opportunities for things to simply go wrong.

Gradually, we are turning an old philosophical doctrine into a reality. We are creating a panpsychic world.

Sigh. No, we're not.

All the computers with which we surround ourselves are starting to be endowed with a rudimentary sentience.

On the other hand, there might be something to that, after all. The easiest way to shop for something online, these days, is to talk about it near my phone. Within an hour I'll have fifty different ads for the product. Like, I'll say something like, "I'm going to bed now," and I wake up and there's ads for Sealy clogging my screen.

Which is not to say I have a problem with targeted ads in general. I'd rather get ads for beer, movies, and cat food (all relevant to me) than for hemorrhoid cream, tampons, or dog biscuits (all utterly useless to me).

We are placing minds everywhere and instilling seemingly inanimate objects with mental experience.

Are we, though? I'm sure several philosophers would disagree that these devices necessarily have "mental experience."

By dispersing intelligent artifacts, humanity is awakening the material world.

Oh, sure, and because of that, my grass curses my ancestors every time I have it mowed.

Proponents make three main arguments. The first is that there doesn’t seem to be any principled way to draw the line between conscious and non-conscious. If we are conscious, why not a dog? A paramecium? A protein molecule? A proton? These systems lie on a continuum with no obvious break.

Just because there's no obvious break doesn't mean there isn't a line there. Just because we haven't figured out where it is doesn't mean it doesn't exist.

Second, panpsychism would solve the hard problem of consciousness.

And love can be reduced to chemical reactions in the mind. So?

Third, several of today’s leading theories of consciousness imply panpsychism.

Before Einstein, leading theories of cosmology implied that the Universe was contained within a few thousand light-years, and said nothing about being unable to accelerate past the speed of light.

When debating panpsychism, the question is not whether, but when. Either the world already is panpsychic or it will be.

As usual, binary ideas like this are misleading. Even if we end up building gadgets that could be called "conscious," they're not going to imbue the rock I stubbed my toe on yesterday with consciousness. I kinda wish they would, because then the rock might appreciate the words I had for it.

The books paint a fascinating picture of a fully sentient world. People can telepathically communicate not just with friends and family, but with atoms, burbling brooks, and the planet as a whole. If your friends come over for dinner, the group forms a temporary collective mind that you can commune with. Every act becomes a negotiation: You had better apologize to the brook for urinating on its bank and talk nice to your hand tools. Say the right words to the right atoms and you can heal wounds or fly like Superman. On the downside, villains can brainwash atoms, unraveling the fabric of reality.

And you get to hear the screams of everything you eat. Yes, even the plants. Especially the plants. That's more horror than science fiction.

So anyway, despite my snark, the article's a good read; I don't have to agree with something to appreciate the arguments.

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