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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 2, 2024 at 9:36am

#1061745

Harder to Swallow

Especially for everyone who has "eat less food" on their resolutions scroll, here's a food article.

15 Foods That Weren't Designed For What We Use Them For

Oddly, the URL for the site shows "Foods we don't use as intended," while the headline is basically the reverse. I wonder why that is. On the surface, the two phrases are semantically equivalent, but I wonder if there's some dark psychology at work there, because I'm paranoid. Like, maybe one of those phrases produces an emotional reaction more desirable to the website's advertisers (who I can't see because I'm running an ad-blocker—unless the listed products are actually paid ads, which I wouldn't put past them).

Perhaps the URL version seems too judgemental compared to the headline. Or maybe I'm reading too much into this, but I think it's important to notice word and phrasing choices.

Anyway, thus ends the "writing" and "media analysis" portion of today's blog entry; let's move on to the article.

As soon as a thing is born into the world, it begins to mutate, changing at a sometimes exponential pace, to the point where it might turn into something else entirely.

This is especially apparent with human larvae.

Take the supposedly innocent graham cracker, for instance.

I think it's pretty widely known, now, that the graham cracker vomited forth from stultifying American Puritanism, intended to help people minimize pleasure and stimulation. Fortunately, now, it's just another cookie. They reiterate this in the list itself; I'm only highlighting select items here.

1. Corn Flakes

Dr. John Harvey Kellogg, who invented Corn Flakes with his brother William, did not care for sin, excess, or meat.

So, the same perverted mindset that produced the graham cracker.

2. Dr Pepper

Interestingly, Dr Pepper was not initially envisioned as a carbonated beverage but rather as a syrup to be mixed with carbonated water at the soda fountain.

That's not really an "unintended use." The company itself switched to bottling.

6. Saccharin

Saccharin, an artificial sweetener discovered in 1879 by Constantine Fahlberg, a chemist working at Johns Hopkins University, evolved from a laboratory discovery to a widely used sugar substitute.

Okay, fine, sure, discovered by accident, like many things. But once discovered, it became a sweetening agent, and a sweetening agent it doth remain. Again, not an unintended use. I'm starting to think I'm not seeing any ads because this whole page is an ad.

8. Popsicles

As the story goes, on a very cold day in 1905, little Frank Epperson was mixing soda powder with water out on his porch when he was suddenly called to dinner by his mother. Or perhaps it was bed. The point is, he forgot his concoction outside overnight only to find it frozen solid the next morning. Luckily, the stick he'd been using to stir it was still stuck in the ice, and when he pulled it out, the popsicle came with it, revealing a glorious new invention.

That origin story is faker and less believable than Spider-Man's.

9. Worcestershire sauce

This tasty condiment with a complex flavor profile owes its origin to a fortuitous experiment carried out in the early 19th century by chemists John Lea and William Perrins of Worcester, England.

Leading to generations of Americans being utterly unable to pronounce the stuff.

12. Baking soda

Before there was baking soda, or baking powder for that matter, bread and cake makers had to resort to the use of yeast to make their creations rise.

Ooookaaaay, now I'm sure this is one big ad. Church and Dwight (the makers of Arm&Hammer) have been pushing alternative uses for baking soda on us for decades. Some of them even work.

But baking soda soon went above and beyond its application as a leavening agent. It found use as a household cleaner owing to its abrasive nature and deodorizing properties.

The fridge-deodorant thing, however, probably does not.

15. Aspartame

This artificial sweetener emerged from the laboratory of one James M. Schlatter in 1965. Schlatter accidentally discovered its sweet taste while working on an anti-ulcer drug. A compound he was synthesizing got on his fingers and he tasted it, as you do in a lab full of potentially toxic and definitely mysterious substances.

So... same origin story as saccharin.

Aspartame has now found its way into a myriad of products, including diet sodas, sugar-free desserts, and chewing gum, offering consumers a guilt-free way to enjoy sweetness — aside from that possible cancer risk.

At this point, if they can't replicate studies showing it can cause cancer, despite really desperately trying (studies often conducted and/or paid for by those with a vested interest in the matter, like, say, sugar companies), the only reasonable conclusion is that its cancer risk is no higher than baseline. And even if there were a small risk, you'd have to weigh that (pun intended) against the health problems associated with too much sugar/HFCS/whatever.

Or you could just eat nothing but unsweetened graham crackers, I suppose. Can't call that living, though.

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