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.

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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 7, 2023 at 11:52am

#1059079

International Merlot Day

Inspired by "Invalid Item"

, today is...

International Merlot Day

I know I talk more about beer in here, but let's not forget, I'm a wine snob, too. There was a time, before craft beer started really taking off in the 90s, when I would always choose wine over beer, if only because the "beer" available was that rice-water swill from the Big Brands.

One reason I don't write more about wine is that, while I enjoy it, I don't have the language for it. I don't mean French; I know some French (at least enough to know that Merlot is pronounced like mare-low and not murr-lott). I mean things like "this dry wine features delicate notes of plum and clover on the nose, and produces a deceptively tart tingle on the palate, where overtones of cherry and fig predominate; the lingering finish is subtly redolent of spices."

I just can't.

It's okay to say that you detect almost any fruit flavor in a wine... except grape. I imagine we can make an exception for wines made from other fruits. One definition of wine is fermented beverage made from fruit; in practice, this is usually grape, but can be any fruit. But in the case of most wines, made from grapes, "grape" is utterly off-limits in the tasting notes.

So I won't describe the difference between Merlot and other red wines, except to say that it tastes almost completely unlike grape.

The most widely-known reds include Merlot, Cabernet Sauvingon, Cabernet Franc, Syrah/Shiraz, and Malbec. They tend to get blended together a lot, too, which is an art in itself. There are, obviously, a bunch (pun intended) more, but those are ones I find at a lot of wineries, and many of them come from the internationally-famous Bordeaux region of France.

But I've had better Merlot right here in the US. This is not to rag on France; I'm guessing that, like most grape-growing regions, they keep the really good stuff for themselves. One day, I'll find out for myself.

Like California. I wasn't overly impressed by California wines until I actually went there and did wine tastings. I'm still not overly impressed with a lot of their beverages; as with beer, there's a rough inverse correlation between how big the producer is (i.e., how many liters they spit out in a year) and how objectively tasty their offerings are. It's not a perfect correlation; you also get real dogs from small vineyards.

And Napa is severely overrated. Seriously. Napa is for tourists. Or, as the bumper sticker here in Virginia asserts: "Napa makes auto parts. Virginia makes wine."

One big difference between wine and beer, apart from what agricultural product they're made from, is that, with very few exceptions, beer has a much narrower range of prices. You rarely, if ever, see a bottle of beer going for thousands of dollars. Not so with wine; I've seen bottles as high as $30,000—in Vegas, where they're obviously catering to jackpot winners and expense-account holders. They also use that as a decoy price; after seeing a $30K bottle, that $100 for a $20 bottle of Merlot seems almost reasonable.

But Merlot is a very common grape, found in many regions that aren't France, as well as being, by some accounts anyway, the most widely-grown grape in its home country. Lots of winemakers in my area produce it. In other words, economically speaking, there's no shortage of supply. So there's no bloody excuse for paying more than around $20 for a bottle; anything more than that, you're just showing off.

And I'll always pick it over Cabernet Sauvignon, but when it comes to reds, well, give me an Argentinian Malbec or an Australian Shiraz.

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