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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QPdoll

about 13 hours ago
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I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 16 hours ago
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Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
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last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
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Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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September 14, 2023 at 9:14am

#1055730

How To Lose Money

Having lived through the Great Coke Crisis of 1985, I'm still fascinated by how companies can sometimes shoot themselves in the foot.

Oh, Schlitz: How a Historic Ad Campaign Helped Kill America’s Biggest Beer Brand

Coca-Cola, of course, survived their mistake (or perhaps it was a calculated conspiracy

). But can we really blame marketing for taking the L out of Schlitz?

Throughout the first half of the 20th century, the Milwaukee-based Joseph Schlitz Brewing Company held the gold crown as America’s largest brewer.

Which glosses right over the dark, dreary days of Prohibition, a significant chunk of "the first half of the 20th century." Like other brewers, Schlitz endured through diversification. But that's irrelevant right now.

Then a series of business decisions, including a disastrous ad campaign, dubbed the “Drink Schlitz or I’ll kill you” campaign, precipitated the downfall of America’s biggest beer brand.

That campaign lasted less than three months, coinciding with a tumultuous year for consumerism in general (1977). It's said that "no publicity is bad publicity," but I'm not so sure about that; still, I'm pretty sure other "business decisions" were more proximate. Like how they stopped making beer and started making formulaic adjunct lager like their competition.

By the late 1950s, Schlitz lost its top title to another quintessential American beer brand: Anheuser-Busch.

Calling that "beer" is an insult to beer, much like calling Kraft Singles "cheese" is an insult to cheese.

During the 1970s, in an attempt to cut production costs and keep up with growing demands, Schlitz’s owners decided to shorten the beer’s brewing time by implementing a process called “accelerated batch fermentation.”

This, by itself, should have been enough to kill the brand. But, as with New Coke, Americans will put up with a lot of enshittification.

They also opted to replace its malted barley with a cheaper ingredient, corn syrup, and began experimenting with the use of a silica gel to prevent haze once the beer was chilled.

That first bit, there, was enough to make it not-beer. Not necessarily bad; lots of great beverages start with corn syrup. But not beer.

And nowadays, of course, lots of people love hazy beers.

Sales dropped as Schlitz’s customers grew frustrated with the brand and started returning cases of beer.

As tasteless as many Americans are, you mess with their favorites at your own peril. (The silica gel thing resulted in a recall, incidentally, never a great thing for any business).

In an effort to stem its declining sales and improve its spiraling reputation, the company hired an ad agency, Leo Burnett & Co., to launch four television spots.

Now, here's the thing: that could have worked. Marketing, done right, can sell anything, including pet rocks and bottled water. You can have the shittiest product in existence, and if you have the right marketing, it'll sell anyway. And truly great products, without marketing, can fade into oblivion.

But by the time Schlitz did those ads, it had already become a terrible product.

Schlitz closed its Milwaukee brewery in 1981.

For context, home brewing became legalized in October of 1978, yet more proof that Jimmy Carter was a severely underrated President. That paved the way for smaller breweries to get some market niches. And Sierra Nevada opened in 1981, still producing craft beer today (I'm not a fan, but I can't argue that it's not real beer).

Today, of course, is a wonderful time to be a beer snob. I'd call it a "Golden Age," but Golden is associated with Coors, and they suck too.

But there are still beer marketing controversies, as I'm sure you remember from earlier this year.

I mention these things sometimes, because even if you're not trying to market anything (a book, perhaps), we're all affected by marketing every day, and a little cynicism can't hurt. And maybe this will help keep you from getting stranded up Schlitz creek without a paddle.

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