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🎶

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
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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.

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Nov 9, 2024 at 11:04am
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Turkey DrumStik

Nov 8, 2024 at 9:47am
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Nov 7, 2024 at 10:19pm
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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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May 13, 2024 at 8:01am

#1071005

The Breakfast and the Breakfurious

"Mommy, where do Pop-Tarts come from?" "The supermarket, kid."

The Contentious History of the Pop-Tart
In the 1960s, two cereal giants raced to develop a toaster pastry

In September 1964, Kellogg’s changed breakfast forever by introducing Pop-Tarts to the world.

Yeah, it sure did change forever. Now instead of an unhealthy breakfast, we can eat a prepackaged unhealthy breakfast.

What made Pop-Tarts so innovative wasn’t just the sweet filling in various flavors squished between two thin pastry crusts. Or that they could be eaten toasted or cold.

I mean, sure, technically, they can be eaten cold, just like leftover pizza technically can be eaten cold. If you're a savage.

It was the convenience with which adults and children alike could open and instantly devour them.

It's not like they didn't have packaged prepared convenience foods in the 1960s. It's just that maybe PTs were the first ones to be marketed as breakfast.

Pop-Tarts’ ingredients mean that they don’t need to be refrigerated, and their foil packaging ensures they can be stored for months.

That's a funny way to phrase "Pop-Tarts contain preservatives." Obviously, most packaged food products contain preservatives. I'm not one of those ooh-booga-booga "all preservatives are bad" people, but it's entirely possible that some are worse than others.

The ingredients are right there

on the package, and most of them are pretty straightforward: sugar, corn syrup, mirror-universe sugar, high fructose corn syrup, etc. But one of them is just called TBHQ.

Now, I'm also not one of those "only eat things you can pronounce" people. As I've noted before, first, it encourages ignorance; second, I did recreational chemistry as a kid; my father had a degree in the field, and I learned how to pronounce lots of things. But I feel like calling it TBHQ deliberately hides a fell secret; it's short for tertiary butylhydroquinone,

which I'm sure if spelled out would freak out your average shopper way more than MSG (monsodium glutamate).

All of which is to say that even I, a big fan of both convenience and better living through chemistry, have my limits.

Throughout the 1950s and 1960s, hyper-sweetened food products that could be eaten on the go exploded in popularity, especially among children.

This just in: children love sweet things. Who knew?

This is all the more ironic because Will Keith Kellogg and John Harvey Kellogg, two brothers from Battle Creek, Michigan, were initially invested in providing healthy foods and cereals to improve digestion when launching their company in 1906.

I suppose you can call that irony. I call it a natural progression. Anything that starts out healthy eventually gets mass-marketed as candy. Cereal products are just one example. Also see: coffee, yogurt.

Just to be clear: I'm not anti-Pop-Tarts. I just like to know what stuff's made of and make my own decisions.

Upon his death on February 10, 2024, William Post was widely identified as leading the team that created the Pop-Tart. Post told southwest Michigan’s Herald-Palladium back in 2003 that Kellogg’s approached him when he was the manager of a Keebler Foods plant in Grand Rapids, where they asked him to develop the revolutionary breakfast food.

On the official Pop-Tarts website there’s no mention of Post.

Of course there isn't. That would be like if Coca-Cola hired some guy named John Pepsi to develop a new soft drink, or if Wal-Mart tasked Betty Amazon to design their new stores.

In order to spread the word of its creation, Kellogg’s used many television shows to introduce Pop-Tarts throughout the last months of 1964. Advertising appeared on “Beverly Hillbillies,” “My Favorite Martian,” “What’s My Line,” “Huckleberry Hound,” “Yogi Bear,” “Woody Woodpecker,” “Quick Draw McGraw,” “Mighty Mouse” and across daytime television.

Product placement is a legitimate marketing strategy. What makes it sneaky is that even if people go, "Hey, that's product placement," it still works. The entire article I'm linking today, for example, is basically an ad for Seinfeld's movie which, since I'm not getting paid either way, I won't name. Nor do I have any desire to watch it, even though it wouldn't cost me a dime to do so as it's on a streaming service I'm already subscribed to.

Ad or not, though, the historical information is interesting to me. I suspect the product was a part of the childhood experience of most people born around the time I was and, as the article notes, it's not like they're going out of style anytime soon.

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