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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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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').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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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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