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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

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Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Kåre Enga in Udon Thani

about 13 hours ago
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

about 13 hours ago
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

about 14 hours ago
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

a day ago
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

a day ago
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

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 2:58pm
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

September 9, 2021 at 12:03am

#1017070

Fight The Power

Here's an article from 2017, but one that seems more relevant now than in the Before Time.

The Five Universal Laws of Human Stupidity
We underestimate the stupid, and we do so at our own peril.

Normally, I'd link to the original source, but it's paywalled. This is a reprint.

Stupid people, Carlo M. Cipolla explained, share several identifying traits: they are abundant, they are irrational, and they cause problems for others without apparent benefit to themselves, thereby lowering society’s total well-being.

I would be remiss if I didn't note that intelligence isn't the last word on human worth. Half the world's human population possesses below-average intelligence (more or less), and it's possible to be stupid and nice, just as it's possible to be smart and an asshole. A university professor like Cipolla might have had some bias in the matter, though.

The only way a society can avoid being crushed by the burden of its idiots is if the non-stupid work even harder to offset the losses of their stupid brethren.

Wait, you want me to be smart and work hard? Let's not go too far, okay?

Of course, everyone does stupid things from time to time. I think it's a matter of proportion.

Let’s take a look at Cipolla’s five basic laws of human stupidity:

Oh, good. Having "The Five Universal Laws" in the headline and not stating them in the body would be... well... stupid.

Law 1: Always and inevitably everyone underestimates the number of stupid individuals in circulation.

No matter how many idiots you suspect yourself surrounded by, Cipolla wrote, you are invariably lowballing the total.

Yeah, best to assume everyone's an idiot until they show otherwise, and then still be prepared to revise one's opinion of them upon further observation.

But like I said... it's about half the population (depending on what's meant by "average").

Law 2: The probability that a certain person be stupid is independent of any other characteristic of that person.

Cipolla posits stupidity is a variable that remains constant across all populations. Every category one can imagine—gender, race, nationality, education level, income—possesses a fixed percentage of stupid people.

This may seem counterintuitive. Okay, it does seem counterintuitive. But the whole purpose of reasoning, and of science, is to check our intuition, which is usually less reliable than logic.

Law 3. A stupid person is a person who causes losses to another person or to a group of persons while himself deriving no gain and even possibly incurring losses.

Cipolla called this one the Golden Law of stupidity. A stupid person, according to the economist, is one who causes problems for others without any clear benefit to himself.

I'm just going to come right out and say it: anyone who trusts horse dewormer over a proven (yes, proven) preventative vaccine is, beyond any shadow of a doubt, in this group.

But to me, the important thing to know is the spectrum this economics professor proposed to map humanity:

This law also introduces three other phenotypes that Cipolla says co-exist alongside stupidity. First there is the intelligent person, whose actions benefit both himself and others. Then there is the bandit, who benefits himself at others’ expense. And lastly there is the helpless person, whose actions enrich others at his own expense.

You'll have to click on the link to see the graphic.

Law 4: Non-stupid people always underestimate the damaging power of stupid individuals. In particular non-stupid people constantly forget that at all times and places and under any circumstances to deal and/or associate with stupid people always turns out to be a costly mistake.

We underestimate the stupid, and we do so at our own peril.

There's a quote attributed to Robert A. Heinlein that's relevant and I've kept it in mind for lo these many years: "Never underestimate the power of human stupidity."

Law 5: A stupid person is the most dangerous type of person.

If "stupid" is defined as above, then yeah, I can believe that.

Declining societies have the same percentage of stupid people as successful ones. But they also have high percentages of helpless people and, Cipolla writes, “an alarming proliferation of the bandits with overtones of stupidity.”

“Such change in the composition of the non-stupid population inevitably strengthens the destructive power of the [stupid] fraction and makes decline a certainty,” Cipolla concludes. “And the country goes to Hell.”

So, based on this, the only way to keep the stupid in check is to actively increase one's position on the article's graph, up and to the right. Can it be done? I don't know. I'm too lazy to try, which I guess puts me in the "stupid" quadrant.

I think it takes an economics professor to think of things in this way. I have to wonder what his thoughts on Rational Market Theory

were, considering the percentage of people who act irrationally. It was in vogue around the time he wrote the above. But I can't be arsed to look it up.

8 Like • Comment • View 5 comments

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