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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 12 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 12 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 12 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 23 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 21, 2020 at 12:00am

#1000587

Halfway Out of the Dark

Today's entry title is shamelessly stolen from the Doctor Who special, "A Christmas Carol."

"30-Day Blogging Challenge ON HIATUS" [13+]:
21. Winter Solstice
Our shortest day and longest night...
We’re about to enter the season of winter: Quiet. Reflection. Incubation. Going Inward.
Write something inspired by that.

"JAFBG" [XGC]
I had neighbors decorating for Christmas before Halloween this year. How do you feel about Christmas starting sooner each year?

Like I wasn't going to talk about the solstice today. The actual event takes place at 5:02 am

Eastern time (which is also WDC time) today, so I'm writing this pre-solstice.

But first: that prompt represents the last of the JAFBG December prompts for me. And I think I've made my feelings known full well about Christmas creep, that phenomenon where Santa Claus sticks his bulbous nose into earlier and earlier months every year.

Except... maybe it's just me, but in recent years, at least in the Before Time, it seems as if more people are getting on the anti-Christmas-creep bandwagon with me, as I have not seen many instances of holiday shit in October, September, August... but then, I didn't get out much even before it was inadvisable to do so, so perhaps I did miss something. My neighbors all waited until after Thanksgiving this year to decorate, and the decorations are pretty much tasteful -- except of course for the one lady who insists on dressing her car up like a goddamned reindeer every year.

But hey, like I said before, I don't begrudge anyone their celebration, especially this year when they need it more than ever. As long as I don't have to hear the endless loop of holiday music, I can deal with it.

One of these days, if I can get up the energy to do so, I want to decorate for Halloween on Christmas. See how they like it when the situation is reversed. But I doubt I could ever be arsed. It's all I can do to keep up with standard maintenance, let alone special decorations.

There's good reason for the lights, of course, even if I often find them to be annoying: the winter solstice represents the longest night. This is, as I'm sure you're aware, due to the real reason for the season: axial tilt.

The article in the link I provided above does a fair job explaining the technicalities of it all. But let me dispel a misconception: Yes, the Earth's orbit is elliptical, and our distance to the sun varies. No, this doesn't do anything noticeable to the seasons. In fact, perihelion -- that point in our orbit where we're closest to the sun -- occurs just a couple of weeks after the Northern Hemisphere winter solstice. No, it doesn't get cold because we're further from the sun; it gets cold because there's basically less sunlight above the equator due to axial tilt.

That's a very simplistic explanation, though; there's more involved, what with weather patterns and atmospheric and oceanic circulation. Seasons tend to lag the solstices and equinoxes; here, the shortest day (today) isn't the day with the lowest average temperatures; that generally occurs in late January. This is one reason why we say that winter begins on the solstice.

But it wasn't always measured in that way. Other cultures considered the solstices and equinoxes to be the middle of their respective seasons. By that reckoning, today is midwinter. I prefer to think of the seasons in that manner, because it means that winter ends earlier, and the sooner this crap season ends the happier I am.

However we define seasons, though, the astronomical milestones remain. We humans have been calculating those since at least the dawn of civilization, and perhaps earlier. Stonehenge is only the most famous of the ancient calendar calculators; there were many others, in disparate cultures.

Another common story we tell ourselves is that our ancestors were frightened that the sun, having descended into darkness, would never come back unless we appeased some gods or spirits or whatever. This story has never sat well with me. The rate of change of day/night hours is a thing they would have watched very carefully; it's fastest around the equinoxes and slowest near the solstices. As the sun's descent at meridian slowed, it would have been blindingly obvious (pun intended, as usual) that it wasn't just going to keep going and eventually disappear; hell, they knew the cycle repeated itself every year (it being the very definition of "year"). So, no, ancient cultures weren't trying to convince the sun to come back; they were celebrating its inevitable return.

They weren't any stupider than we are; they just had less knowledge as to why the seasons changed the way they do. Knowledge isn't the same thing as intelligence. Lack of knowledge isn't the same thing as stupidity.

Though, just like today, I'm sure they had their conspiracy theory advocates, and other people eager to take advantage of the ignorant masses. "If you don't do what I say, the sun will never return! Now give me half your grain and a few virgins so I can... um... appease the spirits, yeah, that's it."

Kind of the ancient version of anti-vaxxers or Qanon believers.

But I have to believe that truth -- that facts -- will eventually prevail. Else, what's the point of anything? Well, maybe there is no point, true, but we can make meaning. We're at a dark time now, what with... you know... everything. But there are glimmers of hope, the promise of a new day. Like the winter solstice, writ large upon the entire world. Metaphorically speaking, of course.

But as someone said on Star Trek: Discovery last week:

Even the darkest night will end. And the sun... will rise.

Look at me being all optimistic and shit. Don't worry, I'll soon be back to my usual grumpy self.

10 Like • Comment • View 6 comments

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