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Saturday, Apr. 27, 2024

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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:

Elisa the Bunny Stik

about 2 hours ago
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

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

August 21, 2020 at 12:05am

#991258

Quantum Speculation

I looked over some earlier entries and discovered a trend to rag on Scientific American articles.

Today's, though, I have fewer problems with.

https://blogs.scientificamerican.com/observations/what-does-quantum-theory-actua...

What Does Quantum Theory Actually Tell Us about Reality?
Nearly a century after its founding, physicists and philosophers still don’t know—but they’re working on it

In essence, this seems to be a good introduction to quantum theory, at least at its very basic.

For a demonstration that overturned the great Isaac Newton’s ideas about the nature of light, it was staggeringly simple. It “may be repeated with great ease, wherever the sun shines,” the English physicist Thomas Young told the members of the Royal Society in London in November 1803, describing what is now known as a double-slit experiment, and Young wasn’t being overly melodramatic. He had come up with an elegant and decidedly homespun experiment to show light’s wavelike nature, and in doing so refuted Newton’s theory that light is made of corpuscles, or particles.

It does, however, toss words around without much precision. For starters, Newton developed several "theories," but the particle nature of light was more of an assumption. And I'd argue that it's more of a refinement than a refutation. Photons act like both, depending on context. Lots of other stuff that Newton came up with turned out to need refinement, but the basic math is still sound, especially for everyday experience.

But the birth of quantum physics in the early 1900s made it clear that light is made of tiny, indivisible units, or quanta, of energy, which we call photons. Young’s experiment, when done with single photons or even single particles of matter, such as electrons and neutrons, is a conundrum to behold, raising fundamental questions about the very nature of reality.

The old argument "is light a particle or a wave?" is usually phrased thus. Personally, I think they're using the wrong terms, as usual. As with Newton, "particles" and "waves" are functions of our everyday experience. The question isn't why light has properties of both; the question is why macroscopic effects are one or the other.

Some have even used it to argue that the quantum world is influenced by human consciousness, giving our minds an agency and a place in the ontology of the universe.

All these years and we're still stuck on that nonsense.

The article goes on to describe the double-slit experiment. I remember doing a basic one of those in physics class when I was in college. It's one thing to hear it described; it's another to actually see it (and do the damned math).

The photon is not real in the sense that a plane flying from San Francisco to New York is real.

That's... misleading. Both are "real." One behaves differently than the other, is all.

Werner Heisenberg, among others, interpreted the mathematics to mean that reality doesn’t exist until observed.

Also misleading, and probably not the case. No, reality doesn't require us hairless apes to observe it, or believe in it. Well, to be fair, there is a nonzero chance that the universe sprang into existence, fully formed, complete with a coherent history. If this event also happened to create humans, then there might be something to this. But while that chance is nonzero, it's represented by an unimaginably small number. Otherwise, the evidence suggests that reality existed before we did, and will continue to exist under most foreseeable conditions after we're all gone.

But quantum theory is entirely unclear about what constitutes a “measurement.” It simply postulates that the measuring device must be classical, without defining where such a boundary between the classical and quantum lies, thus leaving the door open for those who think that human consciousness needs to be invoked for collapse. Last May, Henry Stapp and colleagues argued, in this forum, that the double-slit experiment and its modern variants provide evidence that “a conscious observer may be indispensable” to make sense of the quantum realm and that a transpersonal mind underlies the material world.

Sigh. Yeah, I know. I wasn't going to talk about panpsychism again. But if everything is conscious, than anything can be the "observer," not just humans.

On a related note, you've probably heard of the Schrodinger's Cat thought experiment. Look it up if you haven't; I can't be arsed to explain the whole thing. The upshot of it is that Schrodinger postulated that without an observer, a cat in a box subject to the randomness of quantum phenomena is neither alive nor dead, but in a state of superposition, until an ape opens the box and collapses the cat's wave function.

What's always bothered me about this (and as far as I know, no one has actually performed the experiment, much to the relief of half the world's cat population) is that in his scenario, there is one conscious entity who "knows" whether the cat is alive or dead; to wit, the feline in the box.

Also, there are other ways of interpreting the double-slit experiment. Take the de Broglie-Bohm theory, which says that reality is both wave and particle.

There you go throwing words around again. There's an upper limit to where wave/particle duality holds sway. As this article notes, that limit is as yet undetermined. But it's definitely smaller than a golf ball. A golf ball is unquestionably (except to the most drug-addled philosophers, which would seem to be a lot of them) a reality-based object, and it doesn't act like a photon in the double-slit experiment.

Crucially, the theory does not need observers or measurements or a non-material consciousness.

So when drug-addled philosophers start talking about how nothing exists unless we observe it, give them different drugs. Well, who knows. Maybe they're right. But I doubt it.

If nothing else, these experiments are showing that we cannot yet make any claims about the nature of reality, even if the claims are well-motivated mathematically or philosophically.

Decent conclusion, anyway. "We don't know yet" is a great answer in science.

And given that neuroscientists and philosophers of mind don’t agree on the nature of consciousness, claims that it collapses wave functions are premature at best and misleading and wrong at worst.

I'd be a lot more concerned if neuroscientists and philosophers agreed on anything.

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