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 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.

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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.

November 9, 2023 at 10:07am

#1059195

International Tempranillo Day

Hot on the heels of "International Merlot Day"

and "Harvey Wallbanger Day"

, it seems that today is...

International Tempranillo Day

Don't look at me; I don't make these things up. Apparently, it's set for the second Thursday in November every year, which, sadly, means it can never fall on the same day as Merlot Day. I was looking forward to blending them.

I wanted to highlight this observance for several reasons:

1) I like Tempranillo. It's usually a blending variety, but I'm pretty sure I've had pure Tempranillo and liked it;

2) In my discussion of wine regions the other day, I shamefully left out Spain, Portugal, and Italy, where this grape tends to thrive (I also left out Germany, which, with a few exceptions, really should stick to beer);

3) The promoter of International Tempranillo Day is named "Tempranillo Advocates Producers and Amigos Society (TAPAS)," and that made me laugh.

Clearly, TAPAS

is an American organization, because only English speakers would dare to create such a boldly horrible forced acronym, and the UK wouldn't want another war against Spain; Canada doesn't produce much Tempranillo, if any; and Australia and New Zealand are too classy to torture language like that.

As for the grape itself, obviously, it's associated with Spain in the way that Merlot and others are associated with France, but it's a very popular variety, so it's been transplanted to other places, like California. I suspect, though, that more people have heard of Merlot. That's partly because, as I noted up there, it's often used in blends; and partly because France likes to think that good wine can only come from their side of the Pyrenees (and their side of the Alps), and they have a better marketing cartel.

The name, incidentally, refers to it ripening earlier than other grapes (same Latin root as other "time" words like tempo, tempus, temporary, temporal, and tempura). I imagine this is a desirable quality for winemakers, as it means their harvest and production periods (and thus, output) can be extended, if they have other vines planted as well.

As for the wine itself, it's been a while since I've had any, so I couldn't describe the taste even if I could bring myself to use oenophilic language like I did in the Merlot entry. I don't keep wine-tasting notes the way I do for beer, but I do remember that it's quite tasty, though probably not as bold as the common French varietals.

It's also, as I understand it, a common grape used in port.

Port can only be labeled "port" if it comes from Portugal, which, as I'm sure you can guess, has roughly the same wine-growing conditions as Spain, being on the same peninsula and all. I mean, I don't know what penalties they could impose if someone in California called their product "port;" war, I suppose. Trade, shooting, nuclear, or otherwise. This is analogous to how champagne has to come from Champagne, or tequila can only be called tequila if it comes from a small region of Mexico, outside of which it must be labeled mezcal or agave spirits or whatever.

But the point here is that port can be made from many different grape varietals, and blends thereof, but the ubiquity of Tempranillo in the region means it's a popular one for the famous fortified wine.

So there you have it. If you like red wine, give it a try next time you see it (doesn't have to be today); it may surprise you, and that surprise might even be pleasant.

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