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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

about 10 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 10 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 10 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

about 20 hours 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

a day ago
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

a day ago
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

a day ago
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

a day ago
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

a day ago
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?

December 22, 2020 at 12:11am

#1000643

Tunnels Aren't Boring, or Are They?

History is cool and never really boring, unless you approach it like my teachers did in school, with endless memorization of dates that shit happened, without any context. I think it takes some experience and lifelong learning to be able to have a syncretic approach to historical events.

"30-Day Blogging Challenge ON HIATUS" [13+]:
22. Today in History
Look up some events that happened on December 22nd,
select one, and tell us about it.

That is, with the possible exception of December 22nd, which, while Wikipedia lists a whole slew of events that took place

on this day in history, it turns out that nearly every single one of them is yawn-inducing.

For example:

1920 – The GOELRO economic development plan is adopted by the 8th Congress of Soviets of the Russian SFSR.

Who did what to which now?

1965 – In the United Kingdom, a 70 mph speed limit is applied to all rural roads including motorways for the first time.

And so? A dozen years later, the US adopted a 55 mph speed limit, and our roads are straighter, wider, and feature driving on the correct side.

1891 – Asteroid 323 Brucia becomes the first asteroid discovered using photography.

I mean, two of the few subjects that interest me more than history are photography and astronomy, and this is still meh.

1807 – The Embargo Act, forbidding trade with all foreign countries, is passed by the U.S. Congress, at the urging of President Thomas Jefferson.

If there is anything more boring than international trade negotiations, I haven't discovered it.

It is true that, per the link above, some cool shit happened too, like the reopening of the Brandenburg Gate in Berlin, but there really isn't much there for me to want to write about. Even the births and deaths lists are largely boring. Also, can the dates really be trusted before the adoption of the Gregorian calendar?

But if I had to pick one -- and I don't, really, but it's only polite to address the prompt -- I'm going to go with one related to civil engineering.

1937 – The Lincoln Tunnel opens to traffic in New York City.

...which led to a traffic jam that persists to this day.

The Lincoln Tunnel is an approximately 1.5-mile-long (2.4 km) tunnel under the Hudson River, connecting Weehawken, New Jersey, to the west with Midtown Manhattan in New York City to the east.

Can I just point out how much sheer fun it is to say "Weehawken?" I do that every time I go through the Lincoln Tunnel, which may help to explain why I'm single.

Turns out that the 1937 date is just for one of the current three tubes of the tunnel:

Construction of the central tube, which originally lacked sufficient funding due to the Great Depression, started in 1934 and it opened in 1937. The northern tube started construction in 1936, was delayed due to World War II-related material shortages, and opened in 1945. Although the original plans for the Lincoln Tunnel called for two tubes, a third tube to the south of the existing tunnels was planned in 1950 due to high traffic demand on the other two tubes. The third tube started construction in 1954, with the delay attributed to disputes over tunnel approaches, and opened in 1957.

I would have hated to be the guy supervising the construction of the other tubes while the first one was in operation. "Now, don't screw this up or you'll flood the tunnel."

The tolls on each crossing are only collected in the New York-bound direction.

That's because if you charged people to go into New Jersey, no one would go into New Jersey.

As of 2016, both directions of the tunnel carry a combined average of 112,995 vehicular crossings every day.

Mostly at the exact same time I'm trying to cross.

The article goes on to describe the tunnel's specifications and the negotiations leading up to its construction, which I find fascinating but everyone else would probably feel about it the way I feel about trade negotiations, so I'll skip most of the commentary on it.

Officials from the federal, state, and city levels were in attendance at the ceremony on the New York side, where New York City Mayor Fiorello H. La Guardia and New Jersey Governor A. Harry Moore wielded picks to dig up the ceremonial first mound of dirt.

Personally I think it would have been better if the governors had done most of the work themselves. You know, give them something to do besides trade negotiations.

The first tube was formally dedicated on December 21, 1937. The opening ceremony was accompanied by a military parade on the New Jersey side, as well as the detonation of a series of aerial bombs launched from military ships.

'Muricans gonna 'Murica.

The first vehicles began passing through the tube at 4:00 AM the next day.

Look, I know the construction crews most likely drove vehicles through the tunnel before that, but still, imagine being the first family through a new tunnel bore. "Hey, look, Mildred. Is that... a crack? Leaking water?"

Due to the limited capacity of the new tube, heavy trucks were temporarily banned, and a minimum speed limit of 20 miles per hour (32 km/h) was imposed until a second tube could be completed.

The current minimum speed is 0 mph and the max is about 3. Hell, Google usually routes me to the GWB instead. At least that way I get a view of something besides 80-year-old tunnel walls.

5 Like • Comment • View 4 comments

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