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 11 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 13 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 29, 2022 at 12:03am

#1032975

Food History

Once again, Cracked exposes some of the scurrilous lies we've been subjected to our whole lives.

15 Foods That Aren't From Where We Thought

As usual, I'm not copying all of them here; just a few I feel the need to comment on.

Gone are the days when you were stuck eating whatever your grandma learned how to make from her grandma.

Yeah, Grandpa gets a shot now too.

Today, you can host an international culinary gang bang if you want, but in the process of globalizing food, some people became fusion pioneers, tried to pull one over on us, or just plain got confused.

One thing that's always bugged me is "authentic" cuisine. I don't give a ratatouille's patoutille if a food is "authentic." I only care that it tastes good. One of the few perks to living in the declining days of civilization is being able to get food from all over the world, and to mix it all together if you want.

15. Philly Cream Cheese is From New York

When the inventors of cream cheese were trying to come up with a name for their product in 1880, they settled on “Philadelphia cream cheese” because the city was known to be off the chain, dairy-wise, even though everything about the product was distinctly New Yorkian.

I have no idea if this (or many of the things on this list) are verifiably true or not, but come on, now, New York and Philadelphia? It's not like they called it London Cream Cheese.

14.Hawaiian Pizza is Canadian

Everyone’s least favorite flatbread could more accurately be called “Hawaiian-inspired pizza,” as it was created in Ontario by a Greek immigrant inspired by the sweet-and-savory flavors of Americanized Chinese food to capitalize on the growing Tiki trend.

On the flip side, Canada needs to stay in its lane, food-wise. As an aside, if you want to start an internet argument, the only surer way than to call Chicago "pizza" "not pizza" (as I always do) is to come down hard on liking or disliking ham and pineapple pizza.

The fact that it's ham and not Spam should, incidentally, be your first clue that it's not actually Hawaiian pizza.

12. French Fries Are Belgian (And Kind of Spanish)

Potatoes are a distinctly New World food, first brought to Europe by Spanish explorers and sliced and fried in Galicia in the 16th century, but the crispy little salt delivery vehicles didn’t really take off until Spanish colonists brought them to Belgium, which was called the Spanish Netherlands at the time, just to confuse us in the future.

This is debatable. The name "French fries" is almost certainly American, but as the article points out, potatoes themselves are definitely American (at least continentally speaking). At least one notable source claims that fries (chips, whatever) were a French thing.

That link claims (in French) that they were originally called "pommes de terre frites", which is either "fried potatoes" or "scorched earth apples," depending on how well I've been learning French. Today, the French, and French-speaking Belgians, just call them frites (you probably already knew this; it was one of the French words I learned before I seriously started learning French). Still no word on what the English call English muffins. Actually, I'm kidding; from what I understand, they just call them "muffins."

And yet, an Uber driver from Paris whom I've had occasion to ride with (and practice pronunciation on) says that frites are Belgian, so who do you trust? A food historian or an expat Uber driver?

On the other hand, Parisians are like New Yorkers in that they think everything of note was invented in their home city. In some cases, it's even true. So. Like I said. Debatable.

10. Croissants Are Austrian

Case in point.

8. Vindaloo is Portuguese

and

7. Tempura is Also Portuguese

Look, I've got nothing against the Portuguese, but I'm not taking culinary lessons from a culture that's primarily known for salted cod.

5. Egg Rolls Are American

Those Chinese-American restaurateurs also created the egg roll as a version of the lighter and crispier spring roll that would appeal more to their American customers, who we all know hate the lighter versions of things.

Which of course explains why they're no longer making Bud Light. Just kidding. They piss out way too much of that.

And finally (seriously, go to the article for more of this)...

1. No One Wants to Claim Danishes

Legend has it that Danishes were invented by accident by a thoughtless French chef in the 17th century, who opened a bakery in Paris to sell the “thousand leaves” pastries that became so popular that he franchised to Italy, where Austrian tourists apparently noticed them. Fast forward to 1850, when striking Danish bakers were scabbed over by bakers from Vienna, who didn’t know how to make Danish pastries, so they made … Danish pastries. It became a hit in Denmark, where it’s called a “Viennese,” while Germans call it a “Copenhagener” and Americans, of course, a “Danish,” because despite its universal popularity, no one apparently wants to take the credit.

With a history like that, one wonders how Europe ever devolved into World Wars.

Anyway, food is like language: constantly changing and borrowing shit from other countries. While it's of academic interest to me to figure out the origins of certain foods, ultimately, it was all because some primitive proto-human once decided to eat a morsel that accidentally fell into this newfangled "fire" thing. It caught on, and deliberately burning their food enabled humans to more easily digest protein, which in turn enabled us to power our enormous brains, which eventually led to people continuing to be stupid anyway.

Or something like that.

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