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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 10 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 12 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.

March 26, 2022 at 12:01am

#1029484

Ill-Annoying

Speaking of pizza (see yesterday's entry), it's time now for one of the most polarizing arguments in the world, courtesy of Cracked. No, not the pronunciation of ".gif" (it has a hard g, end of argument).

The New York Vs Chicago Pizza Debate - As Explained By A Chicagoan

Not mentioned: why a New Yorker would care about anything someone from Chicago says.

New York vs. Chicago. A rivalry that's pretty lopsided in favor of Chi-city when it comes to basketball and good public transit, but when it comes to food, both cities have real claims of being among the culinary capitals of the world.

And right there is exactly why we should dismiss anything this author has to say out of hand. I don't know or care about basketball, but NYC absolutely has good public transit. If by "good" you can ignore the filth, rats, and panhandlers. Those are all just part of the charm.

And Chicago has no such claim on culinary capitalism.

Both cities are also known for specialty pizzas: New York for massive, foldable slices of thin crust; Chicago for big ol' cast iron-cooked deep dish. And that's the debate that's liable to get some passionate pie lovers throwing fisticuffs in an airport.

I'm not saying that the Chicago deep-dish stuff is bad. I've had some that I've really liked. What I'm asserting, and I will brook no argument on this, is that it is not pizza. You can't just add cheese and tomato sauce to a casserole and call it a pizza. No, what it is, is a pot pie without an upper crust. Calling it pizza is like calling yourself a car when you're lying down in a parking spot.

I mean, just go to the article and look at the pictures.

And hey! Come here. Lean in closer. Shhhhhh, don't tell anyone, but … it doesn't reheat well (unless you want to take 20 minutes to do it in an oven, which you won't). Putting deep dish in the microwave is for people who are at their most hungover or who truly hate themselves – so college students.

Reheating actual pizza in a microwave is an unholy abomination against all that is good and pure in the world. This is not just me being snobby; it was proven by SCIENCE!. See my entry from back in October on the subject of reheating pizza: "What Do You Mean, "Leftover Pizza?""

I went to Loyola University Chicago, and right across the street from the student union was a spot called Carmen's Pizzeria that had a lunch special of two deep dish slices and a soda for $5. Two slices! Absolute heaven, but go ahead and rule out dinner that day.

I mean, that may have been about 15 years ago based on this author's timeline, but that was absolutely not bad for circa 2007. Cheap enough, in fact, that it makes me wonder about fronts for money-laundering.

Does anything I just described imply healthy behavior? No! I was in college! 34-year-old me actually can't remember the last time I had deep dish. It is a *sometimes* food. If a Chicagoan is actually ordering pizza, they're getting tavern-cut.

What's tavern-cut, you ask? A thin crust, usually pretty crunchy, cut into (preferably asymmetrical) squares. Dibs on the corners and at least two middle pieces, by the way.

I occasionally get a tavern-cut pizza delivered here. It is indeed delicious. It does count as pizza.

Chicagoans don't always want a mountain of knife-and-fork pizza; we want easy-to-consume, belly-filling food that doesn't cost much. Deep-dish is a unique experience; tavern-cut is a light snack/street food that is actually the most popular.

And I have no argument with that. It's different from New York Pizza, but it's still pizza.

The "battle" has already been won. The standard for pizza — from Domino's to CPK to Paulie Rizzioli's Gabagool And Mutzarell on the corner of 47th street next to the bodega with the cat that goes "HEY, I'M FELINING OVA' ERE!" or whatever — is thin crust, tomato sauce, cheese. Yet Jon Stewart had a famous rant about deep-dish not being pizza …

Domino's is shit, CPK is California, where they also don't know how to make a fucking pizza, and that's absolutely what a New York City bodega cat would say. I do know that the last time I was in New York (last month), I got to eat something very like the One True Pizza, down in SoHo, and that made my whole day.

Speaking of which, it occurs to me that Chicago should be my next place to visit. In February, I went to NYC, and then at the turn of the month, I went to Los Angeles. Those are, in order, the two most populous cities in the US, and then the third is Chicago. I've never been closer than the suburbs, so I should probably just go.

I guess I'll need a car, first.

Houston is next after that, by the way. I've gone there before, but it's been a few years.

Anthony Bourdain enjoyed deep dish on camera while being unable to bring himself to call it pizza, only to later call it "an abomination." Why do New Yorkers feel the need to defend so fiercely their precious, plain folding slices?

Because they are PIZZA.

As we've established, Chicagoans just want to eat pizza, like, all the goddamn time. But we also like to bloviate about how much our city rules.

That's what happens when you're #3 and don't even have the movie industry to crow about. It's like some comedian once said (can't be arsed to look it up): "Chicago, right? It's like a bunch of New Yorkers got together and said, 'You know, I really like the crime and overcrowding, but it's just not cold enough.'" Or something like that.

Honestly, I pity New Yorkers. Your lives would be so much improved if you had a more open mind about pizza. And a functioning public transit system. And a functioning Streets and Sanitation Department. And a good basketball team.

Yep. Gotta go to Chicago and see this for myself.

Still don't give a shit about basketball, though.

Helpfully, this article lists a few places to go to eat pizza in Chicago. Now I just need to find some brewpubs there (and probably buy a car) and I'm all set.

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