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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

a day 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

a day 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

a day 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

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?

May 11, 2022 at 12:01am

#1032224

Doom and Gloom

Another article that didn't get the memo that the only ones who can fight climate change are us individuals.

Big oil and gas kept a dirty secret for decades. Now they may pay the price

This article is from about a year ago, so I'm going to go out on a limb and guess that they didn't pay the price.

Via an unprecedented wave of lawsuits, America’s petroleum giants face a reckoning for the devastation caused by fossil fuels

And they have an unprecedented army of lawyers, rivaling only that of Disney, to keep them from facing said reckoning.

Coastal cities struggling to keep rising sea levels at bay, midwestern states watching “mega-rains” destroy crops and homes, and fishing communities losing catches to warming waters, are now demanding the oil conglomerates pay damages and take urgent action to reduce further harm from burning fossil fuels.

But I thought the solution was organic, locally-sourced reusable cotton bags.

But, even more strikingly, the nearly two dozen lawsuits are underpinned by accusations that the industry severely aggravated the environmental crisis with a decades-long campaign of lies and deceit to suppress warnings from their own scientists about the impact of fossil fuels on the climate and dupe the American public.

Scientists? Meh. What do they know? Have they built a giant company to slake the public's thirst for energy? No? Then what good are they?

“Things have to get worse for the oil companies,” he added. “Even if they’ve got a pretty good chance of winning the litigation in places, the discovery of pretty clearcut wrong doing – that they knew their product was bad and they were lying to the public – really weakens the industry’s ability to resist legislation and settlements.”

Not going to happen. Corporations, by definition, can do no wrong.

In 1979, an Exxon study said that burning fossil fuels “will cause dramatic environmental effects” in the coming decades.

“The potential problem is great and urgent,” it concluded.

And yet, here we still are, Cassandra.

The urgency of the crisis is not in doubt. A draft United Nations report, leaked last week, warns that the consequences of the climate crisis, including rising seas, intense heat and ecosystem collapse, will fundamentally reshape life on Earth in the coming decades even if fossil fuel emissions are curbed.

But the climate changes all the time. Who are we to think we have the power to change the entire climate? No, better to just let the coastal cities drown like nature intended. So what if a lot of people are displaced? It's their fault for living in places like Miami.

Municipalities such as Imperial Beach, California – the poorest city in San Diego county with a budget less than Exxon chief executive’s annual pay – faces rising waters on three sides without the necessary funding to build protective barriers.

Eh, they're just going to slide into the Pacific come the next big earthquake, anyway.

Farber said cases rooted in claims that the petroleum industry lied have the most promising chance of success.

The only time lying is bad is when a Democrat does it.

Exxon worked alongside Chevron, Shell, BP and smaller oil firms to shift attention away from the growing climate crisis. They funded the industry’s trade body, API, as it drew up a multimillion-dollar plan to ensure that “climate change becomes a non- issue” through disinformation. The plan said “victory will be achieved” when “recognition of uncertainties become part of the ‘conventional wisdom’”.

Conventional wisdom says we should drive and fly more. Gotta keep the economy running.

The legal process is likely to oblige the oil conglomerates to turn over years of internal communications revealing what they knew about climate change, when and how they responded.

Hm. I should have bought stock in shredder manufacturers. Oh well.

Really, there's no need to worry about climate change. Pretty soon the global average temperature will decrease again.

Nuclear winter will have that effect.

Anyway, if it's not obvious, at least from that last bit, my comments here are intended as satire. Still, the time to do something about climate change was 30 years ago. Now, it's too late. I know they keep trying to get us to hope, but it's not working on me. Like I've said before, I'm riding this sucker down with the hot wind in my hair and a big grin on my face.

4 Like • Comment • View 2 comments

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