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Sunday, Nov. 17, 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:

tj-turkey-jobble-jobble-hard-J

about an hour ago
In response to "Revisited: "A Frank Discussion""

I became disgruntled with meat sold in local grocery stores; half the weight is water, the taste is bland, and it's "use by" date is usually one day away. I found a small town butcher/locker plant that sells locally raised meats and buy in bulk so it's actually fresher, cheaper, and better tasting. Recently, I decided to try my hand at making our own sausage as well. It's a bit of work, but I think it's worth the trouble.

1 Like

Turkey DrumStik

a day ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

Nov 14, 2024 at 12:10pm
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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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