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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 19 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 1, 2019 at 12:13am

#963522

Stupid Is As Stupid Does

Welcome to August. This month, I'm not following prompts. This month, I might not blog every day; I'll be spending most of a week in Vegas, which is three time zones away, and between that, windowless casinos, and the inevitable drinking, I won't even know what day or time of day it is.

I'm looking forward to that. But until then, how about some music - and philosophy?

You look into her eyes, it's more than your heart will allow
In August and everything after
You get a little less than you expected, somehow

http://nautil.us/blog/the-case-for-professors-of-stupidity

The Case for Professors of Stupidity

On this past International Holocaust Remembrance Day, I reread a bit of Bertrand Russell. In 1933, dismayed at the Nazification of Germany, the philosopher wrote “The Triumph of Stupidity,” attributing the rise of Adolf Hitler to the organized fervor of stupid and brutal people—two qualities, he noted, that “usually go together.” He went on to make one of his most famous observations, that the “fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt.”

Yeah... that idea reverberates back even further. Consider Yeats:

Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world,
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.

The linked article goes on to relate Russell's quote to the Dunning-Kruger Effect, which has become fairly well-known - but read the article if you're unfamiliar with the idea.

On a related note, I started calling Bitcoin and other cryptocurrencies "Dunning-Krugerrands." This amuses me far more than it probably should.

Anyway...

But what exactly is stupidity? David Krakauer, the President of the Santa Fe Institute, told interviewer Steve Paulson, for Nautilus, stupidity is not simply the opposite of intelligence. “Stupidity is using a rule where adding more data doesn’t improve your chances of getting [a problem] right,” Krakauer said. “In fact, it makes it more likely you’ll get it wrong.” Intelligence, on the other hand, is using a rule that allows you to solve complex problems with simple, elegant solutions. “Stupidity is a very interesting class of phenomena in human history, and it has to do with rule systems that have made it harder for us to arrive at the truth,” he said.

That... well, as usual, I'm no expert, but it sounds contradictory to say that stupidity is more than just the opposite of intelligence, and then define it as (what seems to me to be) the opposite of intelligence. But hey, I can be pretty stupid.

I'd like to take a moment to talk about what else stupidity isn't. It isn't the same thing as ignorance, though the two can be related. Ignorance is not having all the relevant facts. But - who among us can say we have all the relevant facts in a particular situation? No one, that's who. It always pissed me off when someone used the old saw: "Never assume anything, because it makes an ASS out of U and ME." I mean, what the fuck? You have to make certain assumptions. Always, in every case, without exception, or you're paralyzed with indecision. Granted, though, there's no excuse for not finding out all the facts that you can within any time constraints - ignorance can be fixed, but not willful ignorance.

It's no surprise to me that when I see that bullshit quote, it's almost always associated with sports, a profession known for many things - intelligence not among them. Which is not to say there aren't really smart athletes; it's just the old mind / body dichotomy going on.

Intelligence, then, as I see it, is the ability to make beneficial decisions based on this incomplete information; while stupid decisions are the result of failing to make appropriate assumptions.

But, the article makes a good point at the end (and in the headline): we need to study this more. But given the prevalence of what they're calling "stupidity," I doubt that people will accept the results. After all, even those of us who are aware of Dunning-Kruger often have a higher than warranted opinion of our own competence.

The worst thing about the DKE is the conclusion that we don't even know what we're ignorant about, much of the time. I know there have been several instances in my life that I can look back on and go, "Wow, I was being stupid there." Even though it seemed smart at the time. No, I'm not sharing; besides, my mind tends to block these things from my direct consciousness, unless I'm, I dunno, trying to fall asleep and something from 20 years ago pokes me awake.

I'll end with one last chord: it's easy to assume stupidity for other people, while not attributing one's own failings to it. This happens a lot in politics; Democrats assume Republicans are stupid, and vice-versa. Granted, there are a lot of stupid politicians, but we should at least consider the idea that maybe the other side has good reasons - even if based on different assumptions - for what it says and does.

Except maybe on Twitter.

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