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Monday, May. 20, 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/

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:

Aiva Raine

May 17, 2024 at 6:57pm
In response to "Knead Some Dough?"

I grew up on mostly Wonder Bread despite our family having a large dry farm (dry farm being a farm that grows wheat and barley and safflower without irrigation- it relies solely upon mother nature to rain). I was never fond of it and looked forward to the times my mom would laboriously mill and grind the grain we brought back during harvest into flour and make home-made wheat bread. There was and still is nothing better than fresh whole wheat bread with a little butter and honey.

I think we did the Wonder Bread because my mom wasn't a fan of cooking and making bread is a time-consuming process.

I quit doing the white bread decades ago. There are so many kinds of healthy, no HFCS, etc whole grain breads out there that are delicious that I have no idea why any adult would choose the crappy Wonder Bread.

1 Like

Wordsmitty ✍️

May 13, 2024 at 1:18pm
In response to "The Breakfast and the Breakfurious"

And for those who rather not read the Smithsonian article, the History Channel episode that covers Pop Tarts (if you have a cable TV account):

https://www.history.com/shows/the-food-that-built-america/season-4/episode-1

Annette

May 10, 2024 at 12:05pm
In response to "Meditation Clause"

Everytime a stranger asks me to close my eyes, I keep my eyes open to see what they are up to.
Meditation is not for me.

But I think most people would have 45 minutes every day if they took that time away from social media or TV watching.

2 Like

Wordsmitty ✍️

May 9, 2024 at 1:15pm
In response to "Super Stew?"

But . . .

What's the difference between one being in the soup or in a stew? Is one thinner or thicker? How do writers use soup and stew?

1 Like

Elisa the Bunny Stik

May 9, 2024 at 10:11am
In response to "Super Stew?"

"Served in a pot? Maybe in your country."

Honestly, in the US I'd think soup would be served in a pot due to the quantity that ends up being consumed.

1 Like

Lazy Writer est 4/24/2008

May 8, 2024 at 3:02pm
In response to "Be Positive"

Mine always tell me to eat more fruits and veggies.

Annette

May 8, 2024 at 10:52am
In response to "Be Positive"

Asking doctors for advice has its own pitfalls. Because there is no money in recommending a sensible nutrition pattern, they will say things like: Look it up online.

2 Like

🌕 HuntersMoon

May 5, 2024 at 8:50am
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

May 4, 2024 at 5:42pm
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

January 29, 2024 at 11:13am

#1063143

Ice Sea Moons

Mostly just because I find this stuff fascinating, an article from Quanta:

These Moons Are Dark and Frozen. So How Can They Have Oceans?
The moons orbiting Jupiter and Saturn appear to have subsurface oceans — tantalizing targets in the search for life beyond Earth. But it’s not clear why these seas exist at all.

Usual disclaimer here: by "life" they mean probably simple life; complex life is unlikely, and sentient aliens are pretty much off the table. So let's not go imagining little green women.

For most of humankind’s existence, Earth was the only known ocean-draped world, seemingly unlike any other cosmic isle.

Counterpoint: we barely knew other worlds existed, what the stars and planets really were, and certainly not that they had satellites.

Now Earth’s oceans are no longer unique. They’re just strange. They exist on our planet’s sunlit surface, while the seas of the outer solar system are tucked beneath ice and bathed in darkness. And these subterranean liquid oceans seem to be the rule for our solar system, not the exception. In addition to Europa and Enceladus, other moons with ice-covered oceans almost certainly exist as well.

Really, it shouldn't be so strange that there's a lot of water out there. Even before they made these discoveries about outer solar system worlds, we knew that comets were largely water (in the form of ice).

These moons have existed in the frosty reaches of our solar system for billions of years — long enough for residual heat from their creation to have escaped into space eons ago. Any subsurface seas should be solid ice by now.

This sort of thing is, of course, what science is for.

These oceanic moons also offer a grander possibility. Temperate, potentially livable oceans could be an inevitable consequence of planet formation. It may not matter how far a planet and its moons are from their star’s nuclear bonfire. And if that’s true, then the number of landscapes we might explore in our search for life beyond Earth is nearly limitless.

I should note that I saved this article to my queue before I started playing Starfield.

“Oceans under icy moons seem weird and improbable,” said Steven Vance, an astrobiologist and geophysicist at NASA’s Jet Propulsion Laboratory.

And yet, defiantly, these alien seas remain liquid.

Side note: Every time these days that I see "defiantly," my assumption is that someone doesn't know how to spell "definitely." This is not the case here.

Now, it might seem, on the surface (pun intended), not so odd that an ice-covered moon would have liquid water underneath. After all, Earth is rocky (though much of those rocks are underwater), and yet has a molten core. Water is molten ice. But it's not that simple; we're talking about globes much smaller than Earth, worlds whose inner heat has had more of a chance to dissipate.

The article goes on to explain some theories that fit the evidence, but I won't go into more detail here. In summary, other factors produce heat, such as tidal friction.

As one of the prime candidates for under-ice life in the solar system is Europa (moon of Jupiter), the only thing I have left to note is that it's a good thing Arthur C. Clarke was writing fiction when he penned: "All these worlds are yours - except Europa. Attempt no landing there." Because telling humans they can't do something is a sure way to get them to do it.

1 Like • Comment • View 3 comments

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