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

February 3, 2023 at 12:01am

#1044108

Rants At A Picnic

My second-biggest gripe here is: "Only four?"

4 Bad Foods We Can Move on From in the Modern World

Because I can think of a lot more than four.

But this is Cracked, so you gotta account for short atten- SQUIRREL!

The sheer variety and volume of food available in the world is enough to make both your eyes and mouth water simultaneously.

Especially if you're one of the millions who can't access it.

Dark humor is like food: not everyone gets it.

And yes, guy whose go-to-move at parties is to shut down casual chats by bringing up the saddest possible parts of the human condition, I do realize that starvation and food security is still a massive problem both in the U.S. and abroad. Can we move on now, and you can resume your job as a professional hitman for conversations?

Nah, I reserve that for the blog. But that's only because no one invites me to parties or casual chats.

Here, then, are four foods that, as a modern civilization, we can finally kick to the curb.

4. Necco Wafers

Congratulations, you've identified something more hated than candy corn.

The world has changed around you, while you’ve been chopping up sticks of sidewalk chalk and wrapping them in the sort of wax paper that feels like it was collected from an Egyptian tomb. We’re two months away from Gushers with LEDs in them, and you’re still trying to sell us a candy most monkeys would spit out.

Stop giving candymakers ideas.

3. Baby Corn

Not to be confused with candy corn, because as hated as candy corn is, at least it has a flavor.

I, like many American Jews, have a deep, insatiable love for Chinese food. Fried pork dumplings are very possibly my favorite food of all time, and if not, they would at the very least make it handily through the primaries.

I'm just leaving this in here to avoid comments/jokes about keeping kosher. It's not a thing for most American Jews.

However, there is one consistent, unwelcome invader in many entrees at Chinese restaurants: Those fucked-up little corns.

Truth.

2. Plain Cheerios

YOU SHUT YOUR WHORE MOUTH.

But the light is dimming for the default Cheerio in the modern world — the new cereals are faster, sweeter and more colorful.

That is exactly what makes Cheerios great.

Not that I eat cereal much anymore, but on those rare occasions that I do (usually as part of a free breakfast at a cheap motel), the default is Cheerios, not Chocolate Frosted Sugar Bombs.

1. Brazil Nuts

Unlike the other items on this list, I have absolutely no opinion about Brazil nuts. But I tend to avoid them because the idiots I knew as a kid had a rude name for them that I shan't perpetuate by noting it here.

I am sure there was a point in the history of the human race where the precious food inside a Brazil nut was needed for survival, but I highly question their continued relevance past the invention of fire. The value proposition of the Brazil nut is like breaking into a safe in order to retrieve a piece of dry toast.

I've never had to actually break into a Brazil nut. But you can't make that argument about Brazil nuts when there are so many other foods that are difficult to access, like coconuts or pomegranates (pretty sure that's a word meaning "apple made of granite").

Not to mention, there are so many other better nuts that are begging you to eat them! You’re telling me you’re putting on the blinders and digging past peanuts, pistachios and cashews just to draw blood trying to access the non-prize that is the meat of the Brazil nut?

Peanuts are barely food, pistachios are too much work (or too expensive if you get the pre-shelled kind), and the only purpose of cashews is to provide a unique shape in the trail mix.

Anyway, as I said, why stop at four? I can think of lots of foods whose time has passed, especially since Cheerios doesn't belong on that list at all.

For example: mass-produced milk chocolate. It's basically edible plastic with just enough cocoa so the food police can let them call it "chocolate." There are many tastier alternatives. Sure, they're probably more expensive, but we're not talking about price.

Your turn. What food would you want to see relegated to the circular file of history? (If you're vegetarian or vegan, saying "animal products" is cheating.)

3 Like • Comment • View 4 comments

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