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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:

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

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?

March 5, 2024 at 11:12am

#1065605

Ignorance is...

Here's an example of finding an article that aligns with my existing worldview but having my doubts.

New Study Links Optimism To Lower Cognitive Abilities

That doesn't mean it's right, of course. But it also doesn't mean it's wrong.

Optimistic thinking, often celebrated in self-help literature as a pathway to happiness, health, and longer life, can also lead to poor decision-making.

This has been clear to me for a while.

Research from the University of Bath shows that excessive optimism is actually associated with lower cognitive skills such as verbal fluency, fluid reasoning, numerical reasoning, and memory. Whereas those high in cognitive ability tend to be both more realistic and pessimistic in their expectations about the future.

Basically: ignorance is bliss.

Well, no, not really, because ignorance is simply not knowing something, which has nothing to do with how smart you are. Everybody's ignorant about lots of things. But stubborn, willful ignorance is another matter entirely... also not necessarily correlated with intelligence.

In any case, I'll note that the passage I just quoted seems to be very careful not to imply causation. Does excessive optimism make a person less clever, or does being an idiot lead to optimism? Or, alternatively, is there something else causing both effects? Much as I want to believe the article, that seems like an important thing to find out.

“This points to the idea that whilst humans may be primed by evolution to expect the best, those high in cognitive ability are more able to override this automatic response when it comes to important decisions. Plans based on overly optimistic beliefs make for poor decisions and are bound to deliver worse outcomes than would realistic beliefs,” Dr Dawson added.

Okay, I'm going to need a reference on that "primed by evolution" thing, hopefully not from bogus evolutionary psychology. Also, why isn't this "override" attributed to evolution?

“Unrealistically optimistic financial expectations can lead to excessive levels of consumption and debt, as well as insufficient savings. It can also lead to excessive business entries and subsequent failures. The chances of starting a successful business are tiny, but optimists always think they have a shot and will start businesses destined to fail,” Dr Dawson said.

But, clearly, they're not all destined to fail (unless you take the really long view that every business will fail eventually, even if it takes a few centuries). If we didn't have people taking a shot at these things, we wouldn't have businesses at all. It's like the baby turtles' march to the sea.

The study took data from a UK survey of over 36,000 households and looked at people’s expectations of their financial well-being and compared them with their actual financial outcomes. The research found that those highest on cognitive ability experienced a 22% increase in the probability of “realism” and a 35 percent decrease in the probability of “extreme optimism”.

For once, I can't complain about the sample size. But there are other obvious yellow flags here. UK only, for starters. How cognitive ability was determined from a mere survey: did they basically ask trivia questions? Even if not, there are major issues concerning the standard IQ tests. And how do we know that they're studying truly independent variables?

“Unrealistic optimism is one of the most pervasive human traits and research has shown people consistently underestimate the negative and accentuate the positive. The concept of ‘positive thinking’ is almost unquestioningly embedded in our culture – and it would be healthy to revisit that belief,” Dr Dawson added

Sounds to me like someone hated the idea of positive thinking and set out to debunk it. I can understand that impulse, as I share it. But, first, if science goes into an investigation wanting a certain outcome, that outcome is more likely (even excluding fraud or other shenanigans). And second, I have serious questions about the methodology.

So I put this in the category of "stuff I want to believe, and which tracks with stuff I've been saying, but can't trust the science."

3 Like • Comment • View 1 comment

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