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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 18 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 7, 2020 at 12:02am

#990157

Lox Pie

The way language evolves and changes over time is pretty cool.

http://nautil.us/blog/the-english-word-that-hasnt-changed-in-sound-or-meaning-in...

The English Word That Hasn’t Changed in Sound or Meaning in 8,000 Years

So linguistics is something I'm interested in and I wish I knew more about it, which is why I read articles like this one -- which is more about what doesn't change, which is equally fascinating.

One of my favorite words is lox,” says Gregory Guy, a professor of linguistics at New York University. There is hardly a more quintessential New York food than a lox bagel—a century-old popular appetizing store, Russ & Daughters, calls it “The Classic.” But Guy, who has lived in the city for the past 17 years, is passionate about lox for a different reason. “The pronunciation in the Proto-Indo-European was probably ‘lox,’ and that’s exactly how it is pronounced in modern English,” he says. “Then, it meant salmon, and now it specifically means ‘smoked salmon.’ It’s really cool that that word hasn’t changed its pronunciation at all in 8,000 years and still refers to a particular fish.”

Well, now I'm all hungry. I haven't had a lox bagel in months. There's a place near me that makes a good one but... well, you know.

So, I've known about Proto-Indo-European, or PIE, for some time. The idea is that a large number of languages originated from one source, a language that is so lost in the mists of time that we don't even know what it was called, or if it had a name at all. But I like pie almost as much as I like lox, so that will do.

In modern English, well over half of all words are borrowed from other languages.

I'd argue that all of our words are borrowed from other languages, but, well, I'm not a linguist.

Analyzing the patterns of change that words undergo, moving from one language to another, showed how to unwind these changes and identify the possible originals.

I can't help but notice the similarities to genetics, where biologists can trace common ancestry through genetic drift. The really obvious difference there is that horizontal gene transfer is rare in organisms with mitochondria in their cells, whereas horizontal meme transfer happens in language all the time. (I'm using "meme" in the original sense proposed by Dawkins.)

The word lox was one of the clues that eventually led linguists to discover who the Proto-Indo-Europeans were, and where they lived.

This is where it starts to get really cool, because it's never been clear, at least to me, where the people who spoke PIE lived, or how their language, in the form of its descendants, came to dominate half the world.

In reconstructed Indo-European, there were words for bear, honey, oak tree, and snow, and, which is also important, no words for palm tree, elephant, lion, or zebra. Based on evidence like that, linguists reconstructed what their homeland was. The only possible geographic location turned out to be in a narrow band between Eastern Europe and the Black Sea where animals, trees, and insects matched the ancient Indo-European words.

No reason to name something that you don't have in your area. But that alone is a hypothesis that requires further investigation. So they did further investigation.

In the 1950s, archaeological discoveries backed up this theory with remnants of an ancient culture that existed in that region from 6,000 to 8,000 years ago. Those people used to build kurgans, burial mountains, that archaeologists excavated to study cultural remains. In that process, scholars not only learned more about the Proto-Indo-Europeans but also why they were able to migrate across Europe and Asia.

And to summarize the rest of it: horses and the wheel made this culture extremely mobile, so they were able to spread themselves, or at least their language, far and wide.

Which still doesn't explain why you have some very different words for things in languages, like "cheval" in French and "horse" in English. But I suppose they have theories for that, too.

And yet the word for "cat" is so similar across so many different languages that I have to wonder if they brought their feline companions with them as they traveled and/or conquered.

Which would also be pretty cool. After all, someone has to eat all the leftover salmon.

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