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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 19 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 29, 2020 at 12:03am

#991792

Rare Earth

Okay, whoever keeps asking the question, "How could this year get any worse?" -- I want you to stop that shit right now. Never ask that again.

I learned of Chadwick Boseman's death a bit over an hour ago, so I'm getting drunk and watching Black Panther. But that doesn't mean I can't do a quick blog entry at my usual time.

Today's ties in with the entry from a couple of days ago, and it's written by a guy I know.

https://www.syfy.com/syfywire/do-aliens-exist-probably-are-they-intelligent-prob...

Do aliens exist? Probably. Are they intelligent? Probably not.

Just to be clear, my own speculation on the subject, as seen in "No, It's Not 36."

, was based on stuff I'd figured out before I saw Plait's article. As you'll see, we are in fairly close agreement on these things.

I've been asked if I think life exists elsewhere, and my answer is always that I do. This is based upon a single fact: Life got a toehold on Earth very rapidly after the planet formed, just a few hundred million years.

Only in cosmology is "a few hundred million years" considered "very rapidly."

Maybe life usually takes billions of years, and we were lucky. We only have our one example of life arising and so it doesn't really tell us anything about how easy it is. All it tells us is that the chances of it happening are not 0.

The chances could be one in billions, trillions, or even more. Or it could be close to 1 in 1. We simply don't know, and we won't until we find other examples of life and study them closely. If it seems unreasonable for the chance to be 1 in a trillion, just remember: your odds of winning the lottery may be 1 in a billion when you buy the ticket, but once you've won, prior odds are irrelevant. Maybe we won the lottery. We'll find out eventually if this year would stop fucking with us.

A new study using a sophisticated form of statistical analysis shows that life is actually quite likely to arise if conditions allow it. Hurray! However, it also shows intelligent life is far less likely. Boo.

Which is fine and all, but any statistical analysis, sophisticated or not, boils down to guesswork in this sort of thing. And yet, I'm inclined to believe it. Perhaps because of confirmation bias. But just because I'm inclined to believe it doesn't make it true.

But they found that when they ran their calculations, changing the prior assumptions strongly influenced their result, giving wildly different answers. That's frustrating.

That also supports my "don't put too much weight on this analysis" suggestion. Phil goes on to explain some of the math involved, but I've already had too much wine to know what to paste and what not to paste here, so you'll have to read the article.

What about intelligence? Assuming life arose quickly, the odds of intelligent life evolving are actually slim. Looking at whether the probability is very close to 0 (meaning extremely rare intelligent life) or 1 (very common), Kipping's work favors the low probability at odds of just 3:2. In other words, it's more likely intelligent life is extremely rare.

Phil uses approximately the same definition of "intelligence" as I do; to wit: the ability to use technology to go into space. Those are the species we'd be most likely to encounter, if they exist. In my own opinion, sure, we could find there's a highly advanced squid-like civilization under the ice of Jupiter's moon Europa, but our own squids are pretty damn clever and they're not building rockets. More, we can't seem to communicate with them.

But if you like to gamble, it implies that the best bet is that life is common, but intelligence is rare.

This study, this article, and my own thoughts on the subject are not new. There's a thing called the Rare Earth Hypothesis, which, in a nutshell, asserts that the particular combination of conditions the Earth has experienced and allowed creatures to evolve who could build rockets is probably rare in the universe.

He hasn't proven anything, because that's not how this kind of probability works. He's just shown that, given these assumptions, life is likely to be common and intelligence rare.

Leaving this here because it's important. As I noted before, I could be wrong and I kinda hope I am. Also, again, it's a really big universe, and lots of things are possible... somewhere.

Anyway, if you have the time and desire, read the article and watch the video embedded in it. There's also apparently a Tweetstorm at the link, but I've arranged my browser to pretend that the blight on the Universe known as Twitter doesn't exist, so I can't see it.

And now, back to getting drunk and watching Black Panther.

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