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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 10, 2024 at 9:25am

#1062150

Material World

More from that internationally renowned science reporting site, Cracked:

5 Alien Materials That Exist Naturally Here on Earth

Naturally, I need to nitpick the headline.

Planet formation is an active area of research, so they might not have the precise sequence or timeline nailed down. But whatever the actual mechanism, it's pretty clear that everything on Earth, and even the Earth itself, came from outer space. We still get literal tons

of material from outer space every day. Everything, including your body and the air you breathe, has an alien origin: it was not of Earth until it was, at which point some of it started getting recycled, mixed, and recombined.

Astronauts and space pirates are exploring the cosmos, in search of unobtanium, transformium and precious red matter.

That multi-source science fiction reference is doubly amusing to me, as I just downloaded the game Starfield yesterday (featuring astronauts and space pirates and valuable materials) and started playing it.

Their quest may be fruitless, since none of those materials really exist.

Well, philosophically, anything we can think of "really exists" in some manner. But no, like the Easter Bunny, unicorns, honest politicians, and (probably) Bigfoot, they're not part of consensus reality.

5. Flammable Ice

Yes, water is a mineral (just one that happens to be liquid at room temperature), and when methane is trapped in water crystals, that’s called flammable ice. It has a second, equally extreme name as well: “fire ice.”

While I generally don't bother fact-checking stuff from a comedy site, this sounded familiar, so I looked it up. Yep, they're talking about {xlink:https://en.wikipedia.org/wiki/Methane_clathratemethane clathrates. They even lifted the image in the article (and put "methane clathrate" in its caption) from that Wikipedia page. In their defense, the photo is credited to USGS, which I'm pretty sure means it's in public domain, or at least freely licensed.

What exactly a clathrate is, I leave to any curious readers to follow appropriate links. It's a more generalized thing, and fascinating, but irrelevant to the article or my commentary.

So, is this some promising new energy source, ready to power our kitchens and our factories? Maybe! Or, maybe the warming oceans will melt the ice and release gigatons of methane, setting off a chain reaction of further global warming to push us ever closer to catastrophe.

Burning methane produces carbon dioxide, so why not both?

4. Snail Teeth

What is the hardest substance in the world?

A congresscritter's head.

Many of you will answer “diamond.”

And many don't understand the difference between material hardness, toughness, and strength. And pure stubbornness, as with my answer above.

We’ve also found a natural composite material harder than diamond, in a most unlikely place: inside the mouths of snails.

Doubt they'll try to make jewelry out of that.

If you think you should now fear these snails, who will cut and smash through your body without mercy, put those worries aside. In reality, your body is so fragile that any animal’s teeth could make short work of it...

As some cats I've known were prone to proving.

3. Glass Bombs

We’re not totally sure where australites came from. Once we dispensed with the volcano theory, we dubbed them extraterrestrial, having come from meteorites, but we don’t think that’s true anymore. Instead, we think an asteroid struck the ground in the distant past, somewhere around 800,000 years ago. It hit so hard that it threw terrestrial material into space. This material then came back, now transformed by its journey and smoothed through reentry.

As I noted above (of course I had a point to my nitpicking), that terrestrial material was actually extraterrestrial material that had been here longer.

2. Spontaneously Combustible Coconuts

...is the name of my heavy metal Jimmy Buffett cover band.

You’ve run into coconuts many times when you’ve been stranded on desert islands or when you bought one from a roadside vendor. And yet the dried flesh of the coconut holds strange properties and is a class 4.2 hazardous substance.

No idea where the joke ends and facts begin here, and can't be arsed to look it up.

1. Fire-Forged Feces

Bird shit has always held fascinating properties, sending us rushing to mine the stuff to exploit it and sometimes sparking actual wars. When we weren’t using it as fertilizer, we were digging out compounds from it to use in creating gunpowder.

Pretty sure those are the same material: phosphorus.

Like everyone, the kestrel poops. Sometimes, it poops above a Russian coal mine, where the blasts of hot gasses transform the guano into something new. It becomes a mineral, which can be colorless, reddish or lilac.

But, sadly, doesn't spontaneously combust.

3 Like • Comment • View 3 comments

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