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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

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?

December 5, 2018 at 12:45am

#946869

It's Alive!

I am fortunate in that, most days, I can set my own sleep schedule. Most people can't, due to work, kids or other obligations.

Previous entries have gone into more detail about this, but left to my own devices as I usually am, I'm basically biphasic, sleeping twice in a 24 hour period.

I've long believed, without any real evidence to support this, that everyone has their own sleep preferences, and that to disturb these sleep cycles leads to more stress. Well, now I might have some evidence.

https://getpocket.com/explore/item/if-you-re-just-not-a-morning-person-science-s...

I've seen some of these assertions before - that humans have a greater-than-24-hour natural cycle, for instance. I recall one study that left people in a windowless, indoor environment without clocks for an extended period, and let them sleep and wake on their own. I think the study concluded the natural cycle is about 24.5 hours. Which is weird, because a) the earth's rotation is gradually slowing down due to tidal friction from the moon, so if anything we should have inherited a less-than-24-hour cycle from our distant ancestors, and b) Mars' day is approximately 24.5 hours, and we're probably not Martians. So I don't know what's up with that; as far as I know, no one does.

What's not controversial is that there are individual differences from the "standard" sleep cycle of 11pm-7am or thereabouts. I was on such a schedule for most of my working life, and it messed with me. The times when I could sleep past 7 were rare, but I always felt that those were the occasions when I got my best sleep. Further, trying to fall asleep around 11pm was a chore for me.

I still have questions, though, such as: what happens when someone with a nonstandard monophasic cycle switches time zones? Say, moving to another longitude. We know that jet lag is a thing, but usually it's overcome in a few days of adjusting one's internal clock to local time. But can this be used to adjust, say, someone who sleeps from 2-10 in one time zone to adapt to an 11-7 schedule in another, without too much stress?

Seasonal variations in daylight hours almost certainly play a role in this as well. Right now where I am, it gets dark around 5; in the summer, during DST, darkness might not occur until 9pm, with sunrise coming earlier (I don't know offhand when sunrise happens, for the obvious reason that I'm asleep at the time). And we know that many people, myself included, "enjoy" seasonal mood swings related to the amount of light we get from the accursed daystar.

But probably the worst thing about being a night owl is suffering the scorn of morning people. I used to vacation in the Outer Banks of North Carolina every year, with a bunch of friends, and I was usually the last one to go to sleep and the last to awaken (I don't believe in keeping strict time schedules on vacation). Inevitably, I'd get up around the crack of noon, and everyone else would be all sarcastic with their "Morning, sunshine" and "It's alive!" It got on my nerves, and then my friends wondered why I'm antisocial.

Now, I could probably fix some of that by kicking the caffeine habit, but I don't really want to. In fact, I tried it a few times when I was working, but all it did was make me grumpier and give me a headache, so these trials never lasted very long. Perhaps now, when I can pretty much set my own schedule, I could try tapering off on the wakey-wakey juice to see what my unmedicated cycle might be like. But honestly, for the same reason that I can, I don't feel a pressing need to do so. I have far more urgent habits to revise, such as eating better and getting more exercise. Fortunately, there's a 24-hour gym nearby. Unfortunately, I can't be arsed to go to it at any time. Too many other things to do.

So I'll just urge everyone here to recognize that sleeping late is in no way a moral failing; it's natural for many people. Leave us alone and can the snark. We'll get up early if there's something to do; otherwise, let us sleep and don't give us shit about it.

4 Like • Comment • View 5 comments

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