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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Write_Mikey_Write!

about 2 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 4 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

3 Like

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.

2 Like

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.

1 Like

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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

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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