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.

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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.

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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.

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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

July 21, 2023 at 11:57am

#1052884

If You Cheese

I do like a "you're wrong about that" article... if the article itself isn't wrong.

Almost everything you know about cheese is wrong
Shredded cheese doesn’t melt as well and other fascinating facts about fromage.

Forcing French for funny fails.

A bag of pre-shredded cheese from the supermarket can really come in handy if you’re in a hurry and looking for a quick fix of salt and fat.

And you're not starting out stellar, here: cheese contains those things, but its purpose is protein. And, of course, flavor.

Unfortunately, many pre-shredded cheeses also come with a dose of cellulose, which is essentially wood pulp.

It's a nature-derived preservative. Would you prefer an "artificial" preservative?

The solution then is to grate the cheese fresh off the block or skip the grating process completely, says Mary Rizzo, owner of The Cheese Traveler shop in Albany, New York.

I mean, sure. But that's work. Not just the shredding part, but the cleanup afterwards. The convenience of pre-shredded cheese is enticing. If you banned shredded cheese, you'd Make America Grate Again.

Here are some other common fixes (or upgrades) you can make to your cheese game...

Shredded cheese snobbery is right up there with pre-chopped garlic snobbery, in my book. I admit it's better, but it's also often too much like work. Some of these other items, though...

Venture beyond processed cheese

Yeah, I've ragged on this in here before. Unfortunately, lots of people have come to believe that Kraft Singles is what cheese is "supposed" to taste like, or perhaps Velveeta, and there is no fixing that attitude.

Don’t always use air-tight plastic for storage

I have to agree with this one, too, but again, you have to balance the benefits with the extra work involved.

Learn why certain cheeses are crunchy

Because they're coated with nuts? Okay, no, the article explains this as a natural product of cheese aging. Which, admittedly, I didn't know the details of. So I learned something, too.

Sample real American cheese

I'm all for local artisanal food (I am, after all, a craft beer and local wine snob), but let's not limit ourselves to America, even if we live in the US. It is good to know that, according to this article, local cheese is finally happening in the US; I hadn't heard much about it.

Choose your pairings wisely

This is the kind of thing that can only come with experience, and perhaps someone else to act as Cheddi Master. Everyone's taste is different, even discounting the wrecked palate that leads some people to prefer process "cheese," and what works for me might not work for you.

But in the end, this is work that is worth it, to me: finding out which cheese pairs best with which wine or beer, or with other foods. It only improves one's enjoyment of life, and isn't that the whole point?

3 Like • Comment • View 1 comment

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