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Sunday, Apr. 28, 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 20 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

November 28, 2021 at 12:01am

#1022439

Life, The Universe, and Everything

If you can write about something, is it really intangible?

PROMPT November 28^th

Hopefully a little change of pace tonight. Write about something intangible: faith, magic, energy, power, or creativity.
Just choose one topic, and write about it.

Well, sure, I suppose it's still intangible if you can write about it. Consider a hole, for example. A hole is defined by its surroundings. You can't meaningfully point at a spot in mid-air and go, "That's a hole." It's a thing, but it's not a thing, at the same time. Thinking about it too hard makes your head hurt.

I guess, though, I'm limited to one of those five topics in the prompt, which is too bad, because I've been mulling over a rant about the abstract concept known as "freedom." I guess that one can wait.

So let's talk about energy.

As far as I can tell, energy has two distinct, separate definitions:

1) The capacity to do work;
2) The capacity to do work.

The first definition is physical. That is, it comes from physics. "Work," in this context, is force exerted over some distance. (Energy is also equivalent to mass, per Einstein's well-known formula). For example, the force required to move your car over a flat surface for, say, a mile, can be computed, and that force is (usually) supplied by energy produced by the car's engine, be it internal combustion or electric. Unless said engine isn't working, in which case it's supplied by your muscles and cuss words.

This kind of energy has several modes, such as: kinetic, potential, thermal, chemical, nuclear, etc. The important thing to know about energy is that it requires a gradient in order to perform the work. Solar energy works because the Earth's surface is cooler than that of the sun. Well, usually; may not apply in Phoenix in July.

Even though it can be measured and quantified, it's still intangible. A hydroelectric dam doesn't only have water, fish, and dead mobsters behind it; it also stores potential energy, though it's not like you can see or touch such energy. When the water passes through a turbine, it does work on the turbine, which converts (some of) that potential energy, by means of kinetic energy, into electrical energy.

The second definition is physical.

In this case, by "physical," I mean related to your body. Isn't English wonderful? And "work," in this case, is what one commonly gets paid to do (or receive some other form of compensation for), such as digging a ditch, writing an essay, or fixing the turbine in a hydroelectric plant.

At times, you may get the feeling of having low energy, and at these times, you just wanna lie in bed and not do any work. In reality, your body continues to chug along, using its (mostly) chemical energy (first definition) to do fun things like keep your organs functioning. When that process stops, it's called "death," and at times of low energy that can look pretty good -- fortunately, though, at those same times, we lack the energy to stop our energy.

Confusingly, you also don't want to play at such times -- "play" being the vernacular opposite of "work." This unfortunately does not have a corresponding definition in physics.

There's an old, really bad joke, which was featured (spoiler alert) in Ghostbusters:Afterlife: "Why should you never trust atoms? Because they make up everything."

Now, I've never let the facts get in the way of a good joke -- or a bad one (especially a bad one). But as the joke was told by a certified nerd, I winced. Not because the joke was so bad (it was) or because it is so old (it is), but because a nerd should know better.

Atoms don't make up everything.

The sun, for example, doesn't have many atoms in it. It's so hot that the electrons are stripped from the nuclei, and what you have is a bunch of free-floating protons/neutrons and electrons. And an "atom" is defined as nucleus plus electrons. Physicists call this phase of matter a "plasma." As the Sun contains the vast majority of the mass in the Solar System (according to Wiki, 99.86%

), atoms don't make up everything.

Also, dark matter isn't atoms (as far as we know), and it makes up something like 85% of the matter in the universe.

But okay, here on Earth, the vast majority of matter is composed of atoms. (If it seems like I'm using "matter" and "mass" interchangeably, it's because I am, because I can.) Still, going back to that equation I mentioned earlier, all of those (roughly) 1.3*10⁵⁰ atoms can, in theory, be converted to an unimaginably large amount of energy.

Doing so, however, would ruin your day. Hell, they'd probably see the explosion in the Andromeda Galaxy oh, two and a half million years or so from now.

The point being that atoms don't make up everything -- but energy does.

So much for being intangible, I guess.

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