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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Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

September 16, 2021 at 12:03am

#1017490

Hyperlocal

This one's about one of my favorite subjects. Yes, it's from Cracked.

5 Confusingly Delicious Foods That Only Exist In A Single Place In The U.S.

One of the few bright spots of living in this world today is that we can get food from practically anywhere. Not everything, sure; some things don't ship well, and there's not enough demand for others. But restaurants from other regional cuisines crop up everywhere, and I don't care what anyone says, this is an unmitigated boon for society.

But then there's stuff like this.

5. Suburban Ohio's Proudly Uncooked Pizza

...the local specialty of Ohio Valley pizza, where the cheese and toppings are added on cold, after the sauce and crust come out of the oven.

Everyone has an opinion about pizza. For me, it's that New York style pizza is pizza, and everything else (yes, including anything in Italy) is, at best, a pale simulation thereof; at worst, as with Chicago style "pizza," it's not pizza at all.

But I make an exception for Ohio.

The pizza there is different from New York style: thinner crust, and sometimes they cut it into (mostly) squares. But it's delicious. Many years ago, I used to find any excuse to go to Ohio just for Massey's Pizza (we can get NY style here in my town, but not Ohio style). Last time I went to one, though, either they'd changed the way it was made, or my tastes had altered -- whichever, it ain't what it used to be.

But this? This, I'd never heard of.

4. Hawaii's Guilty Pleasure Spam Sushi

"Decolonize your bookshelf" is one thing, but decolonizing your plate is impossible: So many foods that feel too normal to even think about happened because of colonialism. Without empires and armies, Italy wouldn't have tomatoes, Ireland wouldn't have potatoes, and the Pacific islands wouldn't have Spam, which is just as important as the other two.

When it comes to food, I don't give a shit about "cultural appropriation."

The most popular Spam dish by a mile, though, is Spam musubi, a kind of Spam sushi that consists of a fried slice of the pink stuff, a big ball of sticky rice, and some nori to wrap them together.

Unlike the pizza, this one I've seen. Note I said "seen." I cannot ever bring myself to eat Spam. No, the one Hawaiian dish I crave is Loco Moco, which is ground beef and fried eggs over rice. It's a week's worth of calories in one dish, and it's worth every pound.

But I can't be arsed to make it myself, so I'll have to wait until I can either go back to Hawaii, or the local Hawaiian place wises up and starts serving it.

3. Cincinnati's Challenging Chocolate-Cinnamon-Cheese Chili

On paper, it should be impossible to make chili weird enough to earn a spot on its list. The whole idea of chili is you do it your way, and say "screw you" to anyone without the taste buds to handle it. Veggies? Meat? Spice? Thickness? Toppings?

Oh, but they neglect to mention that there are people in Texas who will shoot you dead if you put beans in chili.

You can't go wrong ... but you can go to Cincinnati, and that's pretty close.

This doesn't just mean chili.

I don't know if I've had official Skyline chili as the article describes, but I did once try the general chili-pasta combination and... well, it's not bad.

2. A Sloppy Joe That Isn't The Joe You Know

...there's no real reason that a Sloppy Joe needs to be ground beef on a bun. If you feel crushed under the tyrannical boot of Manwich's dominion, then there's a place where you can escape: northern New Jersey, where a sloppy joe is a triple-decker deli sandwich with cold cuts, cheese, coleslaw and Russian dressing.

I've been to New Jersey way more than I've been to Ohio, and I've never seen this. Depending on the cold cuts, though, that could be damn delicious.

You might know similar versions of these sandwiches as a Reuben or a Rachel -- as if those names had any more meaning -- but these Sloppy Joes actually have a way better documented history than the cafeteria kind. According to the Town Hall Deli in South Orange, the deli sandwich is a recreation of a house specialty from Cuba -- specifically, the specialty of a spot in Havana named Sloppy Joe's.

Okay, no. There may be a superficial similarity, but a Reuben is hot. Always, always, always. Only idiots in the South try to serve it cold (by way of fairness, the North can't do grits or biscuits & gravy, so it tends to even out).

1. Dirt, Just Dirt

...you know what? I'm going to let you bask in this one all on your own.

And for once, after linking a food thing here, I find I'm not hungry at all. I mean, I started to feel some pangs in the middle, there, but after that last one, I don't think I'll be eating anything for a while.

Now I need an article on regional liquors and where I can find interesting ones. Eventually I'll be able to road trip again... I hope.

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