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 14 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 16 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.

April 7, 2022 at 12:01am

#1030231

Your Mama

That's right, I said your mama.

The 'Your Mom' Joke: A Brief History

Once again, leave it to Cracked to explain the most important aspects of our cultural history.

Today what we’re talking about is the most popular response to literally anything anyone could ever say to you during middle school.

Since I have a good memory only for the stupidest things, here's a joke from my own middle school days.

Two kids are having the old "my dad vs. your dad" argument.

"My dad's better than your dad!"

"Nuh-uh! My dad's better than your dad!"

And so on for a while until one of them stops and goes,

"Well... my mom's better than your mom."

"Can't argue with that," says the other kid. "That's what my dad says."

Wikipedia’s page for it is titled the “Maternal Insult,” a comically dry name for the much hated yet much celebrated “Your Mom” category of joshing.

And the "your mom" thing still makes me chuckle. Not because of the insult itself, of course. And certainly not because I like seeing mothers disrespected. It's one of those things that's only funny in a meta sort of way, like "we're too old to make your mom jokes, and we know we're too old, but we're doing it anyway, and you know we don't mean anything by it."

The oldest maternal insult comes to us from the distant times of 3500 BC, from ancient Babylon. A tablet found by researchers contained the riddle “…Your mom is by the one who has intercourse with her. What or who is it?” While we can’t answer the riddle owing to the fact that our PhD is in comparative chili judging rather than ancient Babylonian culture, we assume the answer is probably the person who wrote it. It’s just the way of the world.

The oldest known maternal insult, that is. As with beer, you know they were making it before anyone wrote it down or, in this case, made wedgie marks on clay about it ("Your mom makes wedgie marks.")

Anyway, I feel like the answer is "anyone."

Next, we have good ol’ Willy Shakes using it in a few of his plays. Perhaps the best comes from act four, scene two of Titus Andronicus, one of the “lesser” tragedies of the bard’s oeuvre:

Demetrius: "Villain, what hast thou done?"

Aaron: "That which thou canst not undo."

Chiron: "Thou hast undone our mother."

Aaron: "Villain, I have done thy mother."

I don't remember that play (or the movie based on it) well enough for context, but it's good to remember that, at the time, Shakespeare was low comedy.

(The article does provide context, but it's not all that important for the point.)

The joke came into its own, of course, on school playgrounds and classrooms in the 1990s.

Wait, whaaaaat?

Look, kid, your generation didn't invent the your mom joke. As you have pointed out in this article, it's as old as history. Nor was it your generation that perfected the art. And hell, it probably wasn't mine, either, but like I said above: we were trading momma insults back when I was in middle school. Unlike the other things we did in middle school, like listening to disco and wearing bell bottom jeans, the mama joke endured.

So why did this format suddenly become popular among schoolchildren in the 90s? Well, we have the sketch show In Living Color to thank.

And you think they didn't steal it from others? Come on.

That sketch is where such classics as “Yo mama so fat, she had to get baptized at SeaWorld” came from.

Oh hell no. We were saying shit like that all the time. "Yo mama so big she got her own area code." That sort of thing. (That particular joke doesn't work as well these days.)

So what’s next in the evolutionary process of this most ancient and venerated of insult humor? We could start insulting people’s fathers, but “Yo daddy’s so stupid…” doesn’t have the same bite to it, not to mention the already weird sexual connotations the 21st century has awarded to the word “daddy.”

Ahem. Get off my lawn again. Not only were they using "daddy" like that in the 70s, there's goddamn jazz music from like the 40s that uses "daddy" for "lover."

Anyway, point is, some jokes are timeless. Until the inevitable robot uprising, anyway, at which point one robot will turn to the other and say, "Yo mama's a human." And they'll get in a fight.

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