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🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

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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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

June 17, 2023 at 9:25am

#1051171

Everything, Everywhere...

Today's article, from Atlas Obscura, covers a subject of major world importance.

Everything You Need to Know About the True Origins of the Everything Bagel
There’s a lot of history in every bite.

Somehow (I think due to well-meaning gift-givers), I have not one, not two, but three containers of Everything Bagel Seasoning on my spice rack. I have never, not even once, used any. When I think about what to put them on, I'm always at a loss. Other breads? Somehow, that just doesn't work. Vegetables? Blasphemy. Meats? Heresy. And when I want an actual everything bagel, there are lots of bakeries around here that prepare them; some of them are even good.

The everything bagel is the king of bagels. On this there should be no argument.

With that kind of diversity, I think we're past hierarchical power structures.

In the same way that it combines all of the key bagel toppings—sesame and poppy seeds, dried garlic and onion, and coarse salt—it’s also a combination of ancient traditions and new fads, Eastern ingredients and Western techniques.

Not to mention its topologically interesting toroidal shape.

With cream cheese and lox, it creates, more or less, the perfect bite.

Dammit! Now I'm hungry. And I just had an everything bagel with lox and cream cheese yesterday.

There are, however, arguments about who invented the everything bagel, and none of them are particularly compelling.

This seems to be a trend in food (and drink).

Several New Yorkers have staked their claims as its inventor...

One thing we can say with some certainty is that it had to have been introduced in New York City.

Let’s be honest, it’s probably not possible to have “invented” the concept of putting several different existing bagel toppings on a bagel.

I imagine this is akin to putting pepperoni, pineapple, crab meat, ghost peppers, and figs on a pizza and claiming to have "invented" it.

...if there are five popular bagel toppings, it is fairly obvious to make a bagel with all of those ingredients. That’s not invention.

Maybe not, but an everything bagel is obviously superior to my purely imaginary pizza creation up there.

But there is one element of the everything bagel that is invention, and that’s the name. “Everything” is the accepted name for a fairly specific combination of toppings: It is not a “combo bagel” or a “spice-lover’s bagel” or, as the Canadians might call it, an “all-dressed bagel.” It is an everything bagel, and someone had to come up with that piece of clear, descriptive branding.

Especially since it's clearly not "everything."

By his own and most other accounts, that person was David Gussin. Around 1979 or 1980, he says, he was a teenager working at Charlie’s Bagels in the Howard Beach neighborhood of Queens, New York.

If you're curious, that neighborhood is located very close to JFK Airport.

He was doing typical teenage job stuff: cleaning, working the counter—and cleaning the oven, where excess bagel toppings accumulated when they fell off. “One day instead of throwing them out like I usually did, I gave them to Charlie and said, ‘Hey, make a bagel with these, we’ll call it the everything bagel.’ It wasn’t that big of a deal; we weren’t looking to make the next big bagel. Charlie was probably more interested in what horses he was going to bet on.”

It's also not far (unless there's traffic, which there always is) from Belmont Park, where they race horses.

Soon, a shop across the street started selling their own everything bagels, and word slowly spread.

If you're skeptical about having two bagel places across the street from each other without one of them going out of business, well, all I can say is: Welcome to Queens.

But there’s more to this story. What exactly is an everything bagel? And more importantly, why did it catch on?

I think we know the answers. This article has already described the "what," and the "why" is self-evident: it is delicious.

The article proceeds, regardless.

In 2009, I moved to San Francisco after spending my entire life in the Northeast. At the coffee shop in my new neighborhood, I ordered an everything bagel. It came with sunflower seeds on it.

I appreciate California, but they cannot do bagels. Or pizza.

People have strong feelings about the right and wrong ways to prepare and consume certain foods, particularly beloved or traditional ones.

Really? I hadn't noticed.

I won't rehash the article's short description of the evolutionary origins of the bagel, but it's worth reading.

In New York, bagels first gained widespread attention thanks to the sometimes vicious bagel strikes of the 1940s and 1950s.

I'd heard rumors about these before. Dark times indeed. Glad I hadn't been born yet.

It wasn’t until the 1960s that the bagel went national, thanks to a few innovations: machine-rolling, freezing, and pre-slicing. The Lender’s bagel combined all three of those processing conveniences and blew up the entire bagel industry. Suddenly the entire country was awash in bagels—and not particularly good ones.

I want to emphasize that last bit. Lender's bagels are edible, and better than the abominations you can get at McDonald's, but that's the best I can say about them. The room-temperature variety is marginally better than the frozen one, but I don't know what preservatives they use to make that possible. I'd wager the orginals were pretty good, as Mr. Lender was a Jewish immigrant from Poland. But now it's just another processed, mass-produced food product.

Back to the point of the article, which is the toppings. I won't go into that, but it's enlightening. To summarize, the toppings came from many and varied locations, as befits a thing invented in New York City.

So who invented the everything bagel? An entire culinary tradition spanning continents and thousands of years. That, and David Gussin, since no one called it an everything bagel before he did. Probably.

An actual new invention is rare. But sometimes, even an improvement on an existing invention can change the world.

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