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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:

Elisa the Bunny Stik

about 7 hours ago
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
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

Apr 25, 2024 at 3:09am
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

Apr 25, 2024 at 2:58am
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

November 5, 2021 at 12:05am

#1020862

Solutions

Sometimes, a solution does not exist. Most of the time, though, several solutions exist.

PROMPT November 4^th

You arrive at work today with five donuts and five coffees for the people who are scheduled to be there. But you have a problem. Counting you, six people are there, and all of you love coffee and donuts! How do you and your friends resolve this?

1. Math:

You might have sat there in math class, bored, going, "When will I ever use this crap?" Fortunately for you, I did not.

The fair and egalitarian solution is actually quite simple, but it does require doing some math.

Forget the donuts for the moment and focus on the coffee. You have five coffees. You have six people. Ideally, you'd need to divide the coffee into 30 equal parts, and then recombine them into six, but that presents logistical challenges. So here's the proposed solution:

Find another cup. Since this is an office, you can presume there's a cup, or glass, or some sort of receptacle available. Take 1/6 of each coffee and pour it into the empty cup. Assuming a fairly standard (in the US anyway) 12 ounces of coffee, that means two ounces of each. 2 ounces out of 12 leaves 10 in each and, magically, you end up with 10 ounces in the formerly empty cup as well. Fair, because everyone gets the same amount.

Maybe, like me, you suck at determining how much 2 ounces actually is. Well, one standard US tablespoon is half an ounce. Four tablespoons is therefore 2 ounces. Again, this is an office, so some utensils are certainly present; just use a large spoon; it will be close enough. Unfortunately, the office cup is probably going to be a different shape than the coffee shop cups, so it won't be easy to check, visually, if it contains the same amount of disgustinglife-sustaining fluid -- in which case you'll need to borrow one more cup from the office kitchenette. Empty one of the paper cups into that one, then transfer the contents of the first office cup to the empty one. Then you can visually check if the quantities are close enough, then adjust accordingly.

There's probably a more elegant mathematical solution, but can you really expect us to come up with it before the morning coffee?

With the donuts, it's even simpler to cut 1/6 of each one, leaving 5/6 of a donut for five people, and five 1/6ths of a donut for one person. It may not be ideal to eat pieces of a donut instead of one larger piece of a donut, but hey, it tastes the same and it's all going to the same place.

2. You snooze, you lose:

Who the hell shows up unannounced to work, without anyone texting me to bring six coffee/donut combinations instead of five? Shit, ordering six donuts would earn me fewer weird looks than just ordering five, because six is half a dozen and the donut vendor is just going to assume I'm trying to cut back. Probably cheaper on a per-unit basis, too.

Point being, whoever showed up without notice simply doesn't get a coffee or a donut. Maybe next time they'll figure out they have to text me.

3. Drawing Straws:

Classic survival dilemma solution. Take the coffee stirrers that you no doubt got from the coffee shop. Break them all in half. Break one of them in half again. Take five half-stirrers and one quarter-stirrer and hold them in your hand so only the very end shows. Everyone gets to pick one. Whoever's stuck with the short stirrer is shit out of luck for the day. If you want, you can do separate lots for the coffee and for the donuts.

4. In Reality:

I hate coffee. With five other people, I'm sure at least one of them is trying to lose weight and could be convinced to give up their donut to me in exchange for my coffee. I can even promise to buy them some kale later.

Actually, in reality, this wouldn't happen because I'm retired.

5. Screw You Guys, Then:

Didn't text me to pick up an extra coffee and donut? Screw you guys, I'm drinking all the coffee and eating all the donuts myself. That'll teach ya. Should have paid more attention in math class, or you could have come up with the fair solution, yourself.

Astute readers will note that in exactly zero of the above scenarios do I voluntarily decide to give up all the goodies. I've made peace with that, and you can too.

One-Sentence Movie Review: Last Night in Soho:

It is rare to find a movie outside of one of my preferred genres that is so well-made; it's artistic without being pretentious and tells a great story that's suspenseful and twisty but not gimmicky -- other reviewers might hedge their reviews, knowing that someone out there is going to disagree with them, but that's not a problem for me.

Rating: 5/5

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