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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 14 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 17 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 20 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
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

Dec 23, 2024 at 4:55pm
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

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.

2 Like • Comment • View 1 comment

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