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 2 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 4 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.

December 23, 2022 at 12:01am

#1042055

Mad World

Regular readers know I source Cracked a lot. This one, however, is from Mental Floss.

A Brief History of Mad Magazine

Cracked started out as a competitor magazine to Mad. The former's current incarnation is, as we know, an internet humor site with short articles and pictures with funny captions. The latter's current incarnation is... well, it's been a while since I've looked it up, but the internet version was lame as hell when I did.

Such is the nature of competition.

Thanks to satire like The Simpsons and The Daily Show, it’s hard to imagine a time when irreverent humor wasn’t everywhere. But the 1950s were much different. Anti-establishment humor wasn’t part of the mainstream. Not until Mad magazine arrived to poke holes in everything from politics to movies to advertising. And even if you never picked up Mad, you probably know Alfred E. Neuman, its moronic mascot.

At some point, I'll need to do a rant on how "moronic" is perfectly acceptable in print, but "retarded" has been relegated to "the R-word," even though they come from the same place. Today, however, is not that day.

But who came up with Mad? What prompted a lawsuit over Alfred E.? And why did the FBI feel the need to keep a file on a silly humor magazine?

The article, of course, proceeds to investigate these important questions, but I'll take a stab at the last one: "because it posed a danger to the establishment." Free speech, my ass.

Today, comic books are the source material for movies that gross billions of dollars. But in the 1950s, adults generally perceived them as hot dumpster trash that would rot kids’ brains.

Considering some of the people alive today who were kids in the 1950s, it's clear that something rotted their brains. It wasn't comic books, though.

Every so often, older people will latch on to some new thing that "rots kids' brains." Jazz. Comics. Rock and roll. Weed. Video games. That sort of moral panic. For some reason, they never seem to worry too much about things that actually rot brains, such as environmental lead.

How did comics get such a bad rap? While characters like Superman and Batman were viewed with suspicion, adults were really fixated on crime and horror comics like the ones published by EC Comics. Founded by Maxwell Gaines in 1944 and later run by his son William Gaines, EC was the publisher behind grisly titles like Tales From the Crypt and The Vault of Horror.

To be fair, some of those were damn explicit. Still, did the adults at the time forget about the long history of horror stories told to kids to scare them straight? I guess they did, because of environmental lead, I guess.

The first issue of Mad was actually titled Tales Calculated to Drive You Mad and retailed for 10 cents.

You forgot to add "Cheap." That was always below the price of every issue of Mad I ever saw, regardless of price. But that only got added later; a quick internet search didn't tell me when it started, but it wasn't with the first issue.

A satire titled “Superduperman” ran in the fourth issue and was significant for two reasons. Mocking DC’s hero created strong word of mouth among readers, and it also led to DC—then known as National Comics Publications—sending a strongly worded legal letter demanding Mad stop mocking their most popular character. Did Mad comply? It did not. Did Mad get a lot of legal letters from that point forward? It did.

I'd describe that more as parody than satire, but I'm no expert.

I'm skipping a bunch here, but it goes into Congressional hearings and the disputed origin of Alfred E. Neuman.

Alfred E. Neuman might have been the most recognizable personality from Mad, but he wasn’t the only one. Over time, the magazine would introduce some popular recurring features in the magazine as well as writers and artists who developed followings of their own. While Mad referred to them as the Usual Gang of Idiots, they were some of the most talented visual storytellers in the business.

This is not disputable. One of my most prized possessions is a two-volume hardcover set of the complete works of Don Martin, from Mad.

Then after a discussion of the various artists involved, it gets back to legal battles, and this bit, I think, is important:

In the 1960s, songwriter Irving Berlin and others went after Mad in court. Mad had printed a collection of joke lyrics to be sung to the tunes of popular songs, and the music industry felt this was copyright infringement. The United States Court of Appeals disagreed and sided with Mad, saying that parody and satire were deserving of substantial freedom both as entertainment and as a form of literary and social criticism.

This is, in my entirely amateur opinion, one of the most important free-speech decisions ever handed down by the courts. And apparently it was all due to one silly cartoon magazine.

By the early 1970s, Mad had a circulation of over 2 million readers and was increasingly seen as a vital voice in the counterculture movement.

I would argue that underground comics, such as those featuring the work of one R. Crumb, were more important to the counterculture. However, the thing about Mad is that it was completely aboveground, yet still subversive in its own way.

But by the end of the 20th century, pop culture and humor were changing rapidly. Kids who had grown up on Mad were now crafting their own comedy, and the winking satire once exclusive to the magazine could be seen in films like The Naked Gun, shows like Saturday Night Live, and even The Onion, which would eventually bridge the gap between print and online humor.

No acknowledgement of Cracked? That's okay; I took care of that. One must also not discount the impact of National Lampoon. Anyone else remember that, long ago, Mad copied National Lampoon and produced a movie? It was a terrible movie.

I have no idea if Cracked parodied it or not, but Mad itself did.

Mad Magazine’s time as a rite of passage for teens may be over, but there’s no mistaking the impact it had on popular culture. Without Mad, we might never have gotten the Garbage Pail Kids or “Weird” Al Yankovic (who once guest-edited an issue, incidentally).

What is subversive eventually becomes mainstream, and then something else subversive crops up. Iconoclasm is a powerful urge in human nature, as is the urge to suppress it.

No need to be Mad about it.

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