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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August 18, 2022 at 12:02am

#1036623

Orange You Special

Hot on the heels of my citrus-themed Comedy newsletter this week, an article about orange juice. (If you missed the NL, here's a link: "Lemons"

)

How orange juice took over the breakfast table
Orange juice used to be a treat you had to squeeze out yourself. More than a century ago, an overproduction of oranges helped create the morning staple we know and love.

Love? Speak for yourself. I'd rather eat an orange. Curious, I tried to look up whether the fruit or the juice is considered better for health; unsurprisingly, as with so much of nutrition science, I got mixed results. I shan't post the links here; just google "orange or orange juice."

It’s bright, but somewhat boring, and bears the dubious halo of being something good for you. Few of us give it much thought, other than to recall its oft-trumpeted Vitamin C content.

Lots of things have Vitamin C. I've gone months without citrus products, and most of my teeth haven't fallen out. Except in that dream. You know the one.

But processed orange juice as a daily drink, you might be surprised to learn, is a relatively recent arrival. Its present status as a global phenomenon is the creation of 20th-Century marketers, dealing with a whole lot of oranges and nowhere to dump them.

Of course it's marketers. It's always marketers. The same people who convinced us to buy chicken wings, pet rocks, and bottled tap water. Much has been made about how "Evian" spelled backwards is "naive," but that's ridiculous; after all, "star" spelled backwards is "rats," for example. No, I'd be way more concerned that one of the French words for the kitchen sink is "évier."

But I digress.

The fruits were shipped all over and eaten fresh or juiced in the home, producing a delicious honey-coloured elixir. California relied on the navel orange and the Valencia orange,; the latter was the best for juicing.

The navel orange itself is interesting. You know how it's got this little knob inside, by the "navel?" That's a buried other orange. It's apparently the tradeoff from genetic engineering for them to have no seeds. Seeded oranges are annoying; they're too much work.

Citrus itself is quite interesting, genetically speaking, but that's another topic.

Florida, however, grew four varieties, and all of these were decent juice oranges. That meant that when, in 1909, the growers met to deal with a burgeoning problem – a glut of oranges, too many for the market to bear – juicing them, rather than curbing their production, was considered a feasible solution.

For context, in-home refrigerators didn't start to be a thing until like the 1920s or 30s.

Be that as it may, oranges, juiced and otherwise, were the subject of a strenuous advertising campaign by orange interests in the 1920s, when the discovery of vitamins was a current event.

Vitamin: what you do when your friends drop by.

Vitamin C was a perfect reason to consume more oranges. Things really got off the ground when nutrition personality Elmer McCollum popularised a mysterious ailment he said resulted from eating too many "acid-producing" foods, like bread and milk: acidosis.

Right, because no one associates citrus with acid.

In fact, true acidosis, which has a variety of causes, cannot be remedied by eating lettuce and citrus, as McCollum claimed. But that didn’t stop the imagination of the citrus industry from taking advantage of this new fear.

Classic marketing: If demand doesn't exist, create it. I've mentioned my First Rule of Comedy before: "Never let the facts get in the way of a good joke. Or a bad one. Especially a bad one." Well, substitute "marketing campaign" for "joke," and you have the basics of marketing conquered.

The promise of a new way to make juice that could be kept frozen, then reconstituted in people’s homes, prompted them into even more production, however. They ramped up tree planting in the 1940s.

Again, though, nothing in here about the parallel development and adoption of the in-home refrigerator/freezer, essential if you're going to sell orange juice concentrate. You might as well write a story about the rise of the automobile and fail to mention gas stations. Wait, this is the BBC. Petrol stations.

When John McPhee checked into a Florida hotel for a reporting trip more than 50 years ago, he discovered that even in the heartland of oranges, fresh juice was a dim memory.

When you only consume inferior product, you start thinking that it's the best. See also: wasabi, milk chocolate, light beer, dried mashed potatoes.

"Next door was a restaurant, with orange trees, full of fruit, spreading over its parking lot," he wrote in his book Oranges. "I went in for dinner, and, since I would be staying for some time and this was the only restaurant in the neighborhood, I checked on the possibility of fresh juice for breakfast. There were never any requests for fresh orange juice, the waitress explained, apparently unmindful of the one that had just been made. "Fresh is either too sour or too watery or too something," she said. "Frozen is the same every day. People want to know what they’re getting."

Which also helps to explain why tourists go to McDonald's instead of local restaurants.

It had taken a few decades, but with the help of advertising and processing technology, the dumping ground for extra oranges was solidly ensconced as its own product, far outpacing oranges themselves in sales.

And don't misunderstand me: people like what they like, and I understand that, even if I sometimes mock it. The problem is that most people only think they've made their own choices.

They haven't. Ad agencies have.

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