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.

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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.

May 16, 2023 at 9:47am

#1049662

A Frank Discussion

I never sausage a thing.

Which Hot Dog Brand Is Best? A Blind Taste Test of Oscar Mayer, Hebrew National, and More
Because your summer BBQs deserve the best. (Or at the very least, not the worst.)

Dammit! It was sitting right there. They could have said "not the wurst," but no, they had to play it straight.

Do you know the difference between sausages, wieners, frankfurters, and hot dogs? If, like me, you hadn’t ever really thought about it and assumed they were all pretty much the same, I’m thrilled to tell you that you’re wrong.

Because of course we are.

It's been many years since I've actually eaten a hot dog, frankfurter, or weiner; anything requiring a hot dog bun. At home, I got really tired of the mismatch between number of franks and number of buns in their respective packages, and while out, there are other foods that appeal to me more. Not to mention I know what they're made of, but that doesn't stop me from eating breakfast sausages.

So, unlike those two other staples of American haute cuisine, hamburgers and pizza, I don't have a dog in this fight. Pun intended, as usual. I just relished the article and found it amusing.

Regardless of which type of sausage is your favorite, there’s one that screams summer louder than the others: hot dogs. Our staff tasters were sure childhood standbys like Oscar Mayer and Hebrew National would sweep the competition but, as always, the results of our taste tests are full of surprises.

Naturally, I enjoyed hot dogs when I was a kid. Our cylindrical delicacy of choice was, unsurprisingly, the Hebrew National brand. When those were unavailable for whatever reason, the replacement still had to be made of cow, because my mom tried to keep a kosher house as best she could out in the boonies.

To cut down on variables, we boiled and tasted only all-beef hot dogs.

So this is why I picked this article to go into my queue.

And though we offered up buns, ketchup, and mustard, most testers boldly chose to taste their dogs plain.

This makes sense from a pure taste-testing perspective, but out in the wild, you're looking for a whole experience, including bun and condiments. I believe that the right choice of bun influences that experience. Would you taste-test pizza without the crust?

As for condiments, in a taste-test, you at least want them to be consistent across all the samples.

And finally, I know they didn't do this in Chicago, because in Chicago, they track down anyone putting ketchup on a hot dog and run them out of town on a rail.

In the end we blind tasted seven of the most popular brands and judged them on flavor, casing snap, and the satisfying firmness of the meat each bite.

Phrasing!

Of course, for full effect, you'll need to go to the article for details. I'm just highlighting things here.

The Biggest Loser: Oscar Mayer

Quelle surprise. Their dogs are terrible.

Unflinchingly Flaccid: Ball Park

I'm starting to think this author has issues.

Not because they don't like Ball Park. That's normal. It's just, again, phrasing.

Happily Herby: Hebrew National

Here's the thing: it's hard to be objective about food (or drinks) during a taste test. Taste is, well, a matter of taste. Beer, for example, is highly personal; some love *shudder* IPAs, while I prefer darker, less hoppy brews. For colas, Coke will, for me, always be far superior to Pepsi. And Hebrew National is always going to be my Platonic ideal of hot dogs, even if I don't eat them anymore, because it was our go-to brand when I was a kid.

The two that beat it on the list, Sabrett and Nathan's, were only available to Kid Me on trips to New York City. Either they hadn't yet expanded distribution to the rest of the country, or we just didn't get them in our off-the-beaten-path area. In both cases, I always wished it was HN.

So, in the end, you'll have to make your own taste test if you care to determine which is best. Or be like most people and just eat whatever's cheapest; this is why many Americans have no sense of taste.

But you like what you like and it's not my decision.

One final note: there is perennial debate over whether a hot dog, nestled as nature intended in its bun, is a sandwich. I've heard even a Supreme Court justice once weighed in on the matter (RBG, if it matters), though in an unofficial capacity.

This is, ultimately, a categorization problem, like whether the Blue Ridge are actually mountains, or Pluto's planetary status. So there's no official answer. Categories are, in the end, a social construct. However, when you consider that the hot dog is generally served with the split side of the bun facing upwards so that the toppings don't fall out—something you never do, and can never do, with a sandwich—and that the bun itself is always solid at the bottom, barring accidents, and also given its origins as handheld street food, there is only one True Conclusion to which someone can arrive:

A hot dog isn't a sandwich.

It's a taco.

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