Complex Numbers (Book)

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Complex Numbers

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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 21 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

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Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

August 13, 2024 at 9:36am

#1075224

Cracking the Code

I vaguely remember a time before these things became ubiquitous.

The Shockingly Relevant History of the Barcode
Mired in consumer protests, legislative battles, and nearly devastating setbacks, the pioneering technology provides a roadmap for others to survive—and even thrive.

Sadly, "barcode" doesn't refer to a way to get the best deals at drinking establishments.

The hype about new technologies often follows a predictable pattern. We’re told a new technology is about to change the world, and then it’s often labeled a failure when it doesn’t make an immediate impact. Take Google Glass, which promised a bold and slightly dorky vision of the future—but ultimately failed in embarrassing fashion.

Oh, that was nothing compared to the rise and fall (pun intended) of the Segway.

However, some of the most successful technologies in the world followed that same pattern but survived what the research firm Gartner calls the “trough of disillusionment,” and people tend to eventually forget about those initial struggles. Few examples illustrate that pattern more clearly than the modern barcode, a technology that celebrated its 50th birthday in 2023.

In reality, it could only have "celebrated" it if it were sentient. Is it?

Before barcodes became so embedded in our lives that we barely noticed them, they went through a dramatic “trough of disillusionment” filled with major consumer protests, legislative hearings, and moments of near-catastrophic failure.

I also vaguely remember the controversy, the most idiotic of which involved invocations of "Mark of the Beast."

On June 26, 1974, the first UPC barcode was scanned on a pack of Wrigley’s gum at a grocery store in Troy, Ohio.

Yes, a milestone to etch into the monuments of history, right next to the Wright flight and Apollo 11.

Despite years of consulting with various stakeholders, the ad hoc committee had neglected one group in particular: consumers.

And? They never consulted consumers before, and hardly ever did again. See also: New Coke.

The major controversy about barcode adoption focused on an issue that might seem a bit strange in retrospect: item-level pricing. Before barcodes, every item in a grocery store had a price tag. Tagging each item required significant labor, and one of the primary reasons grocery stores developed the UPC barcode was to remove the labor costs required by item-level pricing and instead put product prices on shelves.

Grocery stores might actually be profitable if they didn't have to pay those pesky employees. See also: self-checkout (a technology that, obviously, relies heavily on barcodes).

Early barcodes and computerized checkout systems were massively expensive and cost roughly $250,000 per store to install.

Which, if my memory of 1970s prices and wages is correct (it probably isn't), could have paid the salaries of 10 employees for 10 years, or 1 employee for 100 years.

Barcodes survived the struggles of the 1970s. Consumer backlash mostly disappeared by the turn of the decade, likely in part because Tucker-Foreman—who had become the national face of the protests—left her job as the head of the CFA to become the assistant secretary of agriculture at the U.S. Department of Agriculture.

You might need to read the article for the context on that, but basically, the main opponent of barcodes up and joined the enemy.

As for my own experience, I was just a kid then and didn't do much grocery shopping. Mostly, I remember the issue of Mad that poked fun at barcodes. It was from 1978, and it looked like this.

(I hope that works. If not, just do an image search on "mad magazine barcode cover."

Speaking of comics, though, the barcode backlash extended at least into the 1990s there. Comics publishers such as Marvel and DC printed two covers for a while: one with a barcode and another with a box that would normally contain a barcode, which usually sported some action superhero in silhouette. The latter were shipped to direct sales outlets (comics shops) while the former hit racks at newsstands and grocery stores.

So, I guess, you could always tell who was the serious comics collector from that era: the one with all the non-UPC rectangular cutouts on the covers.

Lately, of course, the ubiquity of the UPC barcode has been supplanted by the even more ubiquitous QR square, which helpfully leads smartphone users to malware.

But the UPC code has one great benefit for me: I like to keep track of what beers I drink, and it's easier to use my beer-tracking app to scan the barcode than to search all the different possible permutations of the beer's name.

And that's why barcodes are actually awesome.

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