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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:

Lazy Writer est 4/24/2008

about 15 hours ago
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

about 18 hours ago
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

a day 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

August 4, 2019 at 12:25am

#963685

Headline Questions Are Usually Answered "No."

I should probably type the words "trigger warning" here, because I know some of my readers have mental health issues. But I also know some of you are more triggered by trigger warnings. Oops, too late. Sorry.

http://nautil.us/issue/45/power/does-depression-have-an-evolutionary-purpose

I want to call bullshit on this. I really do. But it's not my field - far from it. My only experience with psychology was a) one class in biological psychology in college and b) depression diagnosis and unsuccessful treatment thereof.

So I'm throwing this out there.

Now, usually, I mine quotes from the articles I link, but few of them in this one are really worth isolating; I think one needs to read the article to get the idea.

I have problems with how evolutionary "explanations" for this and that and the other thing are generally portrayed. I know I've said this before. My biggest problem with it is somewhat addressed toward the end of the article, but only after the author throws out a lot of speculation about how depression might be advantageous from an evolutionary standpoint.

One problem is this: When it comes to things like, I don't know, skull shape, we have an intermittent fossil record thereof. We can see, based on bones dug up here and there, what human, protohuman, prehuman, and other ape skulls look like and how their general forms changed over time. It's easy, then, to see a general trend toward a larger cranium, and how that could accommodate our relatively large brains. We can also see changes in hand / finger bones, and can make a good inference as to why our hands were evolutionarily advantageous.

When it comes to behavior, though, the fossil record is necessarily mute.

With evolution, not every trait is a survival trait. Some are vestigial or effectively so. Others are incidental. One way evolution works is that incidental traits sometimes end up aiding survival and/or reproduction, so those traits can get passed on. Vestigial traits like - I want to say the appendix, but I've heard that might actually be part of the immune system - whether or not you can wiggle your ears are generally neutral to survival, but might have had some benefit in a distant ancestor.

So, in my uneducated view, depression could just as easily be a byproduct of cognition, a neutral one. How can it be neutral when you take suicide into account? Well, as the article pointed out, most suicide attempts fail. Hence, it doesn't necessarily take a person out of the gene pool. Also I should add that while I've been depressed, I've never been suicidal; I don't know what the stats on that are, but I think the article conflates "depression" and "suicidal ideation," and that pisses me right off.

Not caring if you live or die isn't the same thing as actively seeking to end one's life.

So anyway, smells like bullshit to me. To be fair, whenever someone combines "psychology" with "evolution," my BS detector swings into the red, but I'm sure some studies in that area are legit.

I'm more concerned with this holding out false hope to people, or, conversely, possibly making someone think "Oh, well, it's okay if I off myself, then, because evolution wired me to do that." As the article itself points out, "Even if depression evolved as a useful tool over the eons, that doesn’t make it useful today." You know, like the appendix or ear-wiggling ability.

See, we also evolved to have conscious thoughts, to use science, and to otherwise work to overcome atavistic impulses. I, for one, have had my appendix removed; whatever benefit it might have been providing was suddenly outweighed by its cost to my health. We can make these choices.

Still, the article makes some interesting points and I'm curious to see if this line of speculation goes anywhere.

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