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 an hour 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 3 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.

1 Like

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').

1 Like

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.

November 2, 2023 at 7:36am

#1058615

Stout Day

Something different today, inspired by "Invalid Item"

International Stout Day

I really don't know who comes up with these promotional or commemorative "days," but they amuse me. Sure, some of them are blatantly commercial, but then, most official holidays are, too. Others are icky heartstring-tuggers, but we have a few of those on the official list, too.

The important thing is that, sometimes, there's a day related to fine fermented and/or distilled beverages, such as this one.

First, to be clear, the theme of the day is the beer style. The word "stout" itself, as the above link indicates, initially meant proud or brave, but acquired the meaning of strong... though, even later, it became a less offensive way to call someone fat.

But that was after we started calling certain beers stout for their strength.

Now, sure, you can read the text at the link, which, while I can't vouch for its accuracy, tracks with what I know. And you can also view one of the many entries I've done here talking about stout, such as this one from five years ago: "Stout"

But I have one thing to add: as noted at the title link, "stout" used to refer to any stronger beer. Gradually, though, the style came to refer only to stronger porters. So, to recap: Ale refers to almost any beer that's not a lager (it's mostly a matter of what kind of yeast is used, but that's not important right now). Porter is ale made with dark roasted grain. Stout used to be more intense anything, but now is reserved for more intense porter. But that wasn't enough, so they had to invent the imperial stout. A quick rundown of how the first imperial stout came to be can be found here, in another of my earlier entries: "The Yeast You Can Do"

And now, "imperial" is used to modify any beer style that has a higher than normal alcohol content. Imperial stout, such as one of my all-time favorites, Old Rasputin. Imperial lager. Imperial pumpkin ale. That sort of thing. Amusingly, imperial porter.

Such is the evolution of language, style, and beer.

Sure, these definitions can get a bit fuzzy. Where does porter end and stout begin? What's the ABV cutoff between metric and imperial? (No, regular beer isn't called metric; it's just one of my stupid puns.) Is it possible to have a beer that's both ale and lager? (Yes. Cream ale.) These things are at least as much art and marketing as science.

It just so happens that I have some Old Rasputin waiting for me right now. Perhaps it's time for breakfast.

8 Like • Comment • View 4 comments

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