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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/

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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?

January 25, 2021 at 12:02am

#1002747

Navigation

PROMPT January 25th

Write about something antique or inherited that you own. Who owned it before you? Where did it come from? What’s its story?

Unfortunately, I have quite a few inherited items - unfortunately, that is, because my parents have died.

I've talked about some of them in here before, I know, but I can't really remember most of them. I do remember writing about the barometer.

But I can't recall if I've discussed my dad's sextant

or not. Oh well, what the hell, I just fixed myself a martini, so sextant it is.

These days, of course, sailors have other means of navigation, mostly GPS. I have a vague idea of how that works, having used it myself and looked into the (very interesting) technology behind it. What's most interesting about GPS is that if you don't take general relativity into account, it loses precision remarkably quickly. It absolutely relies on science that people in the 19th century couldn't even have imagined, let alone understood. Well, to be fair, if you took the time to explain it to many of them, they'd get it; we haven't gotten any smarter; we've just increased our understanding and changed our technologies.

Still, for the greater part of the 20th century, they understood the principles, but it wasn't until around the turn of the 21st century that GPS became widely available. So as far as I know, a sextant is something that's only about 20 years behind the times. I could be wrong about this. Martini, remember? And so I can't be arsed to look anything up. Just don't take anything I say here as the absolute truth. In vino veritas, but in gin, whatever.

So a sextant is largely obsolete. I like to think that serious sailors keep one around for emergencies, but from what I understand, it's not very useful without two other items: a chronometer and an ephemerides. And in any case, I'm not a sailor like my dad was, so I don't have any actual use for it.

A chronometer is mostly just a fancy word for clock. When mechanical clocks were invented, they relied on a pendulum, a thing that provided a predictable periodic "tick." These were completely useless at sea, what with all the waves and shit. So the big problem in intercontinental navigation was to invent a chronometer that relied on something other than gravity -- but I'm getting ahead of myself. Let me back up.

But first, I'll tell you what an ephemerides is. It's a table of where a certain heavenly body is expected to be at a certain time. These calculations are fairly complicated, but at the same time straightforward. You could have one for the sun, the moon, Jupiter, or any of the other planets or stars.

Now, backing up.

Navigation requires at least four pieces of data. 1. Latitude. 2. Longitude. 3. Heading. 4. Speed. There are probably others, but... gin. Oh yeah. 5. A freakin' map.

Latitude, at night at sea in the northern hemisphere, is dead easy: 1. Find Polaris, the North Star. 2. Determine the angle between Polaris and the horizon. 3. That angle is your latitude. (Step 2 requires an instrument such as the sextant.)

Finding longitude, on the other hand, is complicated as fuck. You have to know the time, and you have to know the expected location, in the sky, of some star or planet or some such. Knowing the time is where the chronometer comes in; knowing the expected location of a certain point of light is the job of the ephemerides.

Heading and speed are largely irrelevant to this description, so I'm going to drink more gin and ignore them for now.

There.

So. You know your latitude because it's night and you've shot Polaris with the sextant. And now you know the longitude, because you know what time it is (or, rather, what time it is back in London or whereverthehell) and you have star charts so that you can tell the difference between where, say, Sirius would appear in the sky from London and where Sirius looks like to you on the heaving deck of a ship.

The reason you know these things is because you have the sextant to determine the angle between the star (or whatever) and the horizon.

There's also a way to "shoot the sun;" that is, figure this shit out in the daytime. This is above my pay grade, even if I weren't three sheets to the wind right now (that's also a nautical phrase, by the way, in case it wasn't completely obvious).

People talk about a "moral compass," what they use to determine their direction in life. A compass is another important tool in navigation (see: "heading"), but it's not the only important tool. I keep the sextant around for two reasons: because it's a constant reminder of what my father lived for a good part of his life (not most; he was a sailor for about 1/4 of his 90 years), and also because it's a reminder that you always need to know your location. Metaphorically speaking.

I told you the other day that I've been going through all the episodes of Star Trek. Picard kept a sextant in his Ready Room -- and given the utter uselessness of a sextant in interstellar space, I like to think it was for the same reason I keep one: that you should always know where you are.

And where you're going.

8 Like • Comment • View 5 comments

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