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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

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

Show Recent Comments

10 Most Recent Blog Comments:

Kåre Enga in Udon Thani

about 20 hours ago
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

about 20 hours ago
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

about 20 hours ago
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

a day ago
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

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:04pm
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 2:58pm
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

October 8, 2019 at 12:21am

#967431

Devolution

https://hedgehogreview.com/issues/animals-and-us/articles/the-deficient-animal

The Deficient Animal
Only the human species is capable of grasping, analyzing, and interpreting signs as symbols.

So, the idea is that the major thing that distinguishes us from other animals is... metaphor. The idea that something can stand for another thing is at the root of language, I reasoned, and we can also nest metaphors.

I've been saying this for years, but I don't have any credentials in the field, so mine is just another opinion. Still, I do enjoy some confirmation bias from time to time.

The article, of course, takes this a bit further. But the author engages in unnecessary obfuscation through esoteric language, so be warned.

The evolution from anthropoid to human was simply a function of the contingencies of adaptation and survival needs within different environments. This increasingly exclusive focus on biological similarities tended, on the one hand, to fold the human being entirely within the continuum of the animal order and, on the other hand, to minimize, downplay, or ignore altogether the distinguishing characteristics of the human species.

"Simply?" I don't think there's anything "simple" about evolution, except insofar as simplicity leads to complexity, but again - not really my wheelhouse.

This school emerged mostly from the work of German and Dutch biologists, zoologists, philosophers, and social theorists, including Paul Alsberg, Louis Bolk, Max Scheler, Adolf Portmann, Helmuth Plessner, Arnold Gehlen, Konrad Lorenz, and the British-American biologist Ashley Montagu, and has been appropriated and refined in recent decades by Jürgen Habermas, Axel Honneth, Hans Joas, Karl-Siegbert Rehberg, and others.

And the Name-Dropping Award of 2019 goes to... James Davison Hunter!

The consensus among the philosophical anthropologists was that the development of human beings was not simply a result of evolutionary progress, but rather of the inhibition of the evolutionary process.

Translation: we evolved by losing survival traits. As near as I can tell, anyway. Though that's not an "inhibition" but rather a new set of survival traits, I think.

In the late eighteenth century, Johann Gottfried Herder had called the human being “the deficient being”; others, following Herder, described humans as animals “not yet determined,” “unfinished,” “incomplete,” “physiologically premature,” and “organically deficient”—and, therefore, ever malleable. In addition to their unfinished character, humans also have no species-specific natural environment they can call home. Home can be anywhere and everywhere; indeed, humans are capable of adapting to a vast range of environments. They are, as the philosopher Max Scheler put it, “open to the world.”

That plopping sound you just heard was more names being dropped.

In sum, human beings must of necessity make up for their instinctual impoverishment by actively transforming the world to suit their own ends, mastering and re-creating nature rather than merely adapting to it. The means by which human beings do this is by representing the world symbolically, particularly through language, but more broadly through culture itself. Culture is a “second nature.”

And that's where this author goes beyond my simplistic assertion above.

The name (or sign) and the object to which the name attaches (the signified) are mediated by an interpretant. In other words, the meaning of a name or sign is manifested in the interpretation it generates in sign users.

Until, ultimately, you end up studying various academic subjects, leading to expressing basic concepts with four-dollar Latin-root words. This guy, by the way, is a professor at UVA, here in my hometown; and yes, they all write that way. (No, I don't know him personally.)

But we have to address the instances of symbol use in our close cousins:

Other species, most famously chimps, bonobos, gorillas, and other primates, are, in limited ways and experimental contexts, able to use signs as designators, combine words into simple sentences, use simple tools, and even create tools. They can do so in signifying, inferential, and ritual ways. They can also interpret the intentions of members of their own species—for example, intentions to mate, to hunt, to attack, or to eat. But they cannot use signs symbolically, as objects of reflection or analysis.

Only the human species is capable of grasping, analyzing, and interpreting signs as symbols. In other words, only humans are capable of using signs at a meta- or self-referential level.

It helps if said signs and symbols are limited to words of two syllables or fewer, but hey, you do you.

So, basically, if I'm understanding this right, the assertion is that without clear, innate survival advantages (claws, sharp teeth, speed, flight for escape, etc.) we had to have some other, more subtle survival advantage, and that advantage is our ability to transfer meaning from one symbol to another or to a referent. This is the foundation of both language and culture, and this adaptation is what eventually led to us doing shit like building skyscrapers and sending robots to explore Mars.

I may be projecting my own thoughts onto the essay, so if you got something else out of it (apart from a headache from all the professorese), feel free to chime in below.

1 Like • Comment

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