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 15, 2023 at 10:05am

#1059521

Forgive Me, For I Have Zinned

Once more inspired by "Invalid Item"

, today marks another wine celebration.

National Zinfandel Day

When I discussed various reds last week, I failed to mention Zinfandel. There are probably hundreds, if not thousands, of grape varieties out there, and I don't even know all of them, but Zinfandel is kind of a biggie, not something relatively obscure like Mourvedre (one of my favorites).

Zinfandel is a bit of an outlier because it came to the US by way of Austria, though the wine is of Italian origin, and the name is... Croatian? Something like that. Actual zinfandel is bold, deep, unsubtle, and so red it's nearly black. The grapes themselves are extraordinarily dark-skinned.

But then they had to go and make White Zinfandel, which is emblematic of everything that's wrong in the world.

Now, to be somewhat fair, I've heard it's improved since the last time I had the misfortune of sipping it. But I'm still avoiding it on general principles.

"White" wine is, of course, not actually white, any more than my skin color is. It can range from pale, almost clear, to a brilliant off-yellow color. The first offense of white zinfandel is that it's actually a blush, or rosé. I'm not expert enough to know much about the processes involved, but from what I understand, the pale pink color in most rosé wine comes from only brief contact with the darker skins of a grape that's generally pale inside.

The second offense is that it's inoffensive. It's the wine equivalent of white bread, American cheese, and light beer: something seemingly crafted to appeal to the lowest common denominator, and I'm not low nor common nor a denominator.

And finally, the wine I tried when it was all the rage in the States was cloyingly sweet. (As I noted above, that may no longer be the case.) I'm not a hater of sweet wines in general; my first wine was, it should come as no surprise, Manischewitz, a fortified wine; I'm a fan of port and its non-Portuguese imitators; and I'll drink the hell out of icewine. Sweet wines have their place. This one just seemed tailor-made to be a wine for people who don't like wine.

"But, Waltz, doesn't that make it a gateway wine?" Maybe. But it's not like there aren't other fine fermented and/or distilled beverages if you're just looking for something alcoholic, and if you don't like wine, then you don't like it, and that's okay. I know people who simply don't like the taste. WZ seems like it's meant to help non-wine people fit in to the rarefied snobbery of wine culture. In that, it fails, because real wine snobs look down their well-trained noses at WZ quaffers, much as I look down my nose at Bud Light aficionados.

It's like they're trying to appeal to a broader market by making a product that's nothing like other instances of that product. Naked cash grab.

Finally, "white" zinfandel tastes completely unlike the red variety, such that when I finally got around to tasting actual zinfandel, it was a real epiphany. I might actually like it better than Shiraz.

There was a bumper sticker floating around some time ago: "Absolve yourself of white zin."

Indeed.

That said, if you like it, then you like it. Apparently lots of people do, or at least claim to; it still sells. I'm not actually ragging on your sense of taste, only how the winemakers addressed it.

5 Like • Comment • View 3 comments

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