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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 8 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 7, 2021 at 12:05am

#1021039

The Einstein Trap

"Don't believe everything you read on the internet."
-Abraham Lincoln

PROMPT November 7^th

Today's prompt is taken from a book I own. "Great Quotes From Great Leaders", published by Motorola, my employer. This one is from Norman Vincent Peale. "The trouble with most of us, is that we would rather be ruined by praise than saved by criticism." Do you feel this is a valid statement? Tell us why you feel the way you do.

The trouble with quotes is that they give the unwarranted weight of fame to questionable opinions.

Which is not to say I disagree with the quote, but I'll get to that in a minute. Well, okay, more like an hour.

What I mean is... well, consider the following quote:

Life is like riding a bicycle. To keep your balance, you must keep moving.

Sounds kind of simple, right? An almost trite comparison. Think about it, though, and a couple of things come to mind: First, most people who ride bikes are perfectly capable of keeping their balance when they're not on the bike, so you're adding a needless level of complexity to life. Perhaps the reason you're riding the bike is to get to a particular destination, in which case that alone is reason to keep moving; balance is irrelevant to that. Further, however balanced you might be on a bike, there's always the possibility that you'll hit something and wreck, or something will hit you and you'll wreck. These possibilities are greatly diminished if you stay sitting at home instead.

In other words, scratch the surface a bit and it's not really a great metaphor. What gives it weight? What enabled me to find it in a quick search on the internet? Well, it's that the quote is attributed to Albert Einstein.

The point being not that the quote is bad -- it's got a certain cleverness to it -- but that if Al Smith of Poughkeepsie had said it instead of the great genius Albert Einstein, it would never had survived. Oh, someone else might have said it eventually, because it's really fairly obvious (unlike the things Einstein is rightfully famous for, such as the Theories of Relativity, which are absolutely not obvious, but someone else would have worked them out eventually too). But someone famous says something, and boom, it's suddenly received wisdom.

"The trouble with most of us," then, is actually that we attach an unwarranted profundity to random sayings of people we've heard of and admire. I'll call it the Einstein trap.

That said, I have no admiration for Norman Vincent Peale and, in actuality, I'm far less likely to consider anything that he wrote to be useful.

He was, after all, one of the founders, if not the founder, of Toxic Positivity. Almost worse, it's very likely

that he just made everything up, including, ironically, all the quotes and anecdotes of unnamed Great Men in his writing.

But... the best (by which I mean the worst, that is, the most successful) con men don't lie all the time; they sprinkle in just enough truth and/or common sense (which is not the same thing as truth) to persuade people that everything they say is truthful.

So by mentally devaluing the quote simply on the basis of who it's attributed to, I fall into the reverse of the Einstein trap; that is, giving words of figures that I don't like less consideration on the basis of that dislike.

I'll admit that this is just as wrong as the Einstein trap itself, so let's consider the quote by itself, stripped of attribution:

The trouble with most of us, is that we would rather be ruined by praise than saved by criticism.

"The trouble" -- this would seem to imply that it's the only trouble, which obviously it's not. But as a rhetorical device, okay, fine.

"with most of us" -- kind of a weasel phrase. How "most" is "most?" And who's "us?" I'm going to go with "greater than 50%" and "humans in general"

"is that we would rather be ruined by praise than saved by criticism." More rhetoric, in this case the... crap, I don't remember the Greek name for it. All rhetorical devices have Greek names. It's the one where you contrast two different states of being: "ruined" and "saved" in this case.

So, can one be ruined by praise? Yes, I think so. Absolutely. It's like when you show your writing to your family and they go "Yes, dear, this is wonderful," and so you gather up the courage to send it to a publisher, who sends back a note like "This isn't fit to line my birdcage." The family's praise isn't helpful.

And can one be saved by criticism? Oh, I think it's obvious that we can, if it's given in good faith and taken without anger.

So I think the quality of the quote hinges on the word "most." And that's the part I dispute. Where are the psychological studies backing up, or negating, the majority assertion? Did anyone to a poll? Has some grad student made it their thesis?

I think "most of us" certainly feel good when praised. Like dogs, we're just made that way. And criticism can be tough to take, but I know I appreciate it if it's not mean-spirited, and I try to use it to improve. But again, I don't have much evidence to generalize that to others.

So. If he'd said "some" or "many" instead of "most," I would be less inclined to be critical of the quote, regardless of its source.

And that's my criticism.

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