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Saturday, Apr. 27, 2024

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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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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

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

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?

March 19, 2024 at 9:18am

#1066562

Stardust

The equinox will occur later today (or early tomorrow if you live in certain time zones), so it's appropriate that this cosmological link resulted from my random die roll today. Warning: It's from Salon, a source I don't normally use and that might be problematic.

Sure, we're all made of stardust. But what does that really mean?
We often hear that our bodies contain elements from the stars. But how do we know this for sure?

Personally, I think it's the most meaningful, profound insight in all of science. As far as I know, most cultures have an origin story: we were shaped from trees; we appeared out of literal nothing; we were formed from clay; whatever. I use passive voice there because, usually, gods were involved, powerful entities that predated humanity. Some origin stories also attempt to explain where those creator gods came from, but you quickly run into the turtle problem: if Earth is a flat disc resting on the back of a giant turtle, what's holding up the turtle? Another turtle? Well, what's holding up that one? It's turtles all the way down.

Now, you could say, "But science runs into the same problem. If all that exists was created in a Big Bang, how and where did the Big Bang... well... bang?"

This is a fair question, however it's stated. And science doesn't have a definitive answer. The difference between that and creation myths is that science doesn't pretend to have a definitive answer.

In any case, one reason I find it meaningful is that it doesn't just focus on one human tribe, or even on humanity in general, but the entire universe; that is to say, everything. How humans came to be from non-humans is the focus of evolutionary biology, not cosmology. And how different tribes formed is in the purview of, among other disciplines, anthropology. I'll grant that the creation stories our ancestors came up with have a certain poetry to them (and say an awful lot about the cultures involved), but to me, they all read like "Just-So" stories. Or, well, vice-versa. But my point stands.

With that, I won't be quoting too much from the article. That was really the thing I most wanted to say.

As the poets at NASA put it, “from the carbon in our DNA to the calcium in our bones, nearly all of the elements in our bodies were forged in the fiery hearts and death throes of stars.”

Poetry again. It depends what you mean by "nearly all."

By mass, it's mostly oxygen (the "ox" in "Hydrox" from yesterday's entry). By number of atoms, we're mostly hydrogen (the "hydr" in "Hydrox" from yesterday's entry). You might also recognize these elements as the atoms of water, and I'm sure you've heard that we're well over 50% water, so that all tracks, though both hydrogen and oxygen are part of other molecules in our bodies, notably carbon-based ones.

Point being that hydrogen is mostly primordial, not forged in stars. Though an argument could be made that, at the very least, protons are mostly primordial, no matter how many of them are in a given nucleus.

And they’ve been around far longer than we have. Light elements started forming an estimated 14 billion years ago, actually, in the first few minutes after the Big Bang, though others didn’t come around till a few hundred thousand years later when the universe cooled down enough for electrons to stay in orbit around atomic nuclei.

That... doesn't quite mesh with my understanding, though I admit my understanding may be off. An element is defined by the number of protons in its nucleus, irrespective of the number of electrons associated with it. A proton is a hydrogen atom (technically, it's a hydrogen ion, but whatever). So once you have free-roaming protons, you have hydrogen. Therefore, it doesn't matter if a proton (or a bound pair or triad of protons, together with the appropriate number of neutrons) has electrons or not; it's still hydrogen, helium, or lithium, respectively.

In fairness, the article calls out these elements in the next paragraph; I just have quibbles with how the information is phrased.

But you don’t just kill a star and get an entire cupboard of elements suitable for whipping up whatever material good — whether Uranus or, with apologies, your anus — you’re after.

Apology absolutely not accepted. Seventh-planet puns are never funny. Well, almost never.

Apart from that, well, the article delves into the science of it all, and I won't rehash it further. Suffice it to say, for my purposes, that we know, insofar as we can ever truly know anything, that we are indeed stardust. So is your dog. And poison ivy. And plastic and concrete. The good, the bad, and the ugly.

5 Like • Comment • View 6 comments

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