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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:

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

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?

March 12, 2024 at 9:26am

#1066147

Mind Matters

This one's from Quanta, and was probably written by a human.

New Theory Suggests Chatbots Can Understand Text
Far from being “stochastic parrots,” the biggest large language models seem to learn enough skills to understand the words they’re processing.

Difficulty: how do you know anyone understands the words they're processing? I'm sure you think you do, and therefore by extension, other humans do, but can you really know for sure that anyone else is anything more than a biological robot?

Artificial intelligence seems more powerful than ever, with chatbots like Bard and ChatGPT capable of producing uncannily humanlike text.

Article is less than two months old and already outdated; Google changed the name of its AI bot from Bard sometime since then. I don't remember what they changed it to. I liked "Bard," so I simply quit messing around with it. Was I programmed to do that? Absolutely.

Do such models actually understand what they are saying? “Clearly, some people believe they do,” said the AI pioneer Geoff Hinton in a recent conversation with Andrew Ng, “and some people believe they are just stochastic parrots.”

Stochastic Parrots is absolutely going to be the name of my virtual EDM Jimmy Buffet band.

This evocative phrase comes from a 2021 paper co-authored by Emily Bender, a computational linguist at the University of Washington.

I shouldn't do it. I really shouldn't. But I'm going to anyway, because my algorithm requires it:

It suggests that large language models (LLMs) — which form the basis of modern chatbots — generate text only by combining information they have already seen “without any reference to meaning,” the authors wrote, which makes an LLM “a stochastic parrot.”

Seriously, though, I see one important difference: an actual parrot is, like humans, the product of billions of years of evolution. She may not understand the words when you teach her to repeat something like "I'm the product of billions of years of evolution" (nor do many humans), but she, like other living creatures, seems to have her own internal life, a sensory array, and desires (apart from crackers). She seeks out food and water, and possibly companionship. She observes. She may not know that her ancestors were dinosaurs, but she inherited some of their characteristics.

In other words, calling LLMs "stochastic parrots" may be an insult to parrots.

Also, the actual definition of stochastic, via Oxford, is "randomly determined; having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely." Which would seem to apply to most living things.

These models power many of today’s biggest and best chatbots, so Hinton argued that it’s time to determine the extent of what they understand. The question, to him, is more than academic. “So long as we have those differences” of opinion, he said to Ng, “we are not going to be able to come to a consensus about dangers.”

I've been wondering why so many people talk about the "dangers" of AI, but never about the very real and somewhat predictable danger of bringing another human life into the world. Said human could very well become a mass murderer, a rapist, a despotic tyrant, a telemarketer, or any number of despicable things. The only time I hear about the dangers posed by new humans is when someone's ranting about immigration, and that doesn't count.

New research may have intimations of an answer. A theory developed by Sanjeev Arora of Princeton University and Anirudh Goyal, a research scientist at Google DeepMind, suggests that the largest of today’s LLMs are not stochastic parrots.

This theoretical approach, which provides a mathematically provable argument for how and why an LLM can develop so many abilities, has convinced experts like Hinton, and others. And when Arora and his team tested some of its predictions, they found that these models behaved almost exactly as expected.

Maybe it's just the way this was worded, but that seems paradoxical. If an LLM were truly autonomous, independent, understanding.. sentient... then you'd expect the unexpected, no?

“[They] cannot be just mimicking what has been seen in the training data,” said Sébastien Bubeck, a mathematician and computer scientist at Microsoft Research who was not part of the work. “That’s the basic insight.”

Except... we humans also mimic what was in our training data. Sure, some of us can take it apart and put it back together in new ways, build on what's been done before, but for the most part, even our innovations are really just variations on a theme.

These abilities are not an obvious consequence of the way the systems are built and trained. An LLM is a massive artificial neural network, which connects individual artificial neurons.

"Not obvious," you know, unless you've read, like, even a small sampling of science fiction.

Big enough LLMs demonstrate abilities — from solving elementary math problems to answering questions about the goings-on in others’ minds — that smaller models don’t have, even though they are all trained in similar ways.

And that sounds quite a bit similar to what shrinks call the Theory of Mind.

“Where did that [ability] emerge from?” Arora wondered. “And can that emerge from just next-word prediction?”

This is what is meant when people claim that consciousness is an emergent property.

Anyway. The article goes on to describe some of the tests they put the models through, and I can't really comment on the methodology because my training data set doesn't really cover those protocols.

Is it conscious? I don't know. I don't know for sure that you are, and vice-versa.

7 Like • Comment • View 2 comments

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