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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Lazy Writer est 4/24/2008

about 21 hours ago
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

a day ago
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

August 23, 2019 at 12:31am

#964576

Consciousness

So, yeah, I took some time off to indulge vices (other vices than posting stuff here, anyway), but I'm back home now. And I have a difficult article to present today.

http://nautil.us/issue/47/consciousness/is-matter-conscious

Is Matter Conscious?
Why the central problem in neuroscience is mirrored in physics.

I'm of the considered opinion that most headline questions are answered "No."

What is physical matter in and of itself, behind the mathematical structure described by physics?

Math, too, has its limitations. Try as we might, for example, we can't precisely describe the shape of a cloud, or a tree, using math. Attempts to do so often result in something resembling a cloud or a tree, but not a particular cloud or tree.

Modern science has given us good reason to believe that our consciousness is rooted in the physics and chemistry of the brain, as opposed to anything immaterial or transcendental. In order to get a conscious system, all we need is physical matter. Put it together in the right way, as in the brain, and consciousness will appear.

Fair enough. But, again, try as we might, even if we could put everything together in the right place to recreate a human (or other animal), such an object would lack life. Life is still an elusive mystery in many ways, but we know it has to come from life, in an unbroken chain all the way back to the first life - whatever that was. Despite our certainty of our identities as individuals - as well as others' identities as individuals - there's a real, non-metaphysical, connection through time and space to all other life as we know it. That's why it will be so important to discover extraterrestrial life: is it connected in some way to ours, or was there a parallel to our own evolution?

My point is that "put it together in the right way" isn't as easy to do as it is to say.

If we were somehow granted knowledge of every physical detail and pattern in the universe, we would not expect these problems to persist. They would dissolve in the same way the problem of heritability dissolved upon the discovery of the physical details of DNA. But the hard problem of consciousness would seem to persist even given knowledge of every imaginable kind of physical detail.

This sort of thing always bugs me. Is it even possible to have knowledge of every physical detail and pattern in the universe? Magic 8-Ball says "No" - if only due to Heisenberg's Uncertainty Principle. And also because to fully model the universe, you'd need something at least as complex as the universe, which you can't have, because of the definition of "universe." Even if some of the wildly speculative alternative-universe hypotheses are true, you'd then have to model those, and the problem persists.

We take it for granted, however, that physics can in principle tell us everything there is to know about the nature of physical matter.

For previously unknown values of "we," anyway. Because I, and a whole lot of people who are smarter and more knowledgeable than I am, don't accept this premise.

There is already a tradition for connecting problems in physics with the problem of consciousness, namely in the area of quantum theories of consciousness.

Which collides with my personal tradition of dismissing, out of hand, any article that discusses "quantum theories of consciousness."

The article continues by going way over my head with the philosophy. I kind of get the impression that the author is trying very, very hard not to use words like "god" or "deity," instead mincing around the theological implications. But I could be projecting my own bias; I don't know.

Mind you, I'm not saying the article is wrong, or right, or doesn't have decent points. It just seems to me to be another approach to the "god of the gaps" issue - that is, there are always gaps in our knowledge, no matter how much we learn about the physical universe, and at that point you might as well ascribe the unknowns to an entity of great knowledge and/or power. Science cannot disprove the idea; again, the gaps are always there, and you can't prove a negative. It just seems like a cop-out.

Then again, I've thought that maybe the entire universe could be, itself, conscious; I mean, look at a map of the universe

and compare it to a neural network of the human brain

.

But that doesn't mean the two have any similarities beyond the superficial.

1 Like • Comment

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