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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Show Recent Comments

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.

August 1, 2022 at 12:02am

#1035939

That's Mental

Starting off August with a bit of history, courtesy of Cracked...

How Exactly Did 'Mind Readers' In The 1950s Fool Millions?

Apparently, lots of people were easily fooled in the 1950s. UFO sightings, nuclear proliferation and families, and the lie that communism was the greatest threat to our way of life.

But there was also this.

In any case, today we’re talking about mentalists.

You know, speaking of the 50s, rock could have headed in a completely different direction had it gone with "heavy mental" instead of "heavy metal."

In the 50s, there was a mentalist act that was famous throughout the English-speaking world. Sydney and Lesley Piddington were a married couple from Australia, but with a name like “Sydney Piddington” it was only a matter of time before they found fame in the UK.

Um, you know where most of the white people in Australia came from, right?

You see, Sydney and Lesley seemed to have an incredible ability: they could communicate things to each other without speaking, seemingly over vast distances.

As opposed to most married people, whose minds are utterly unknowable to each other. In retrospect, them being a married couple that could read each other's minds should have been the first five red flags.

As their popularity grew, they began to perform over even greater distances, often sponsored by the BBC. In one famous instance, Lesley spoke a phrase taken at random from a book by a panel of judges while under armed guard in the Tower of London while Sydney was broadcasting from some other, unimportant, non-London town. England: a country that conquered most of the known world instead of simply building a second city of their own.

You hear that sound, Cracked? That angry rumbling in the distance, getting louder? That's a mob of Mancunians and a mob of Yorkers. If you're really lucky, they'll notice each other before they get to you and start a new War of the Roses. Otherwise, you're boned.

(Technically, London is one of the smallest cities

in the UK. But even if you consider Greater London, non-London cities in England are, based on my experience, fiercely protective of their status as Not London.)

Underwater. In the sky. Separated by many miles and both watched by live audiences – there was seemingly no explanation for how they were able to do what they did. All of the popular explanations of the time were disproven by the Piddingtons in subsequent shows.

You know, even though there's no such thing as telepathy, I gotta respect the creativity it takes to come up with an act like that. Anyone can learn to do cold readings—I used to be pretty good at that, myself—but setting up a "mentalist" act that fools a lot of people for a long time? Props.

The Piddington’s act was as confounding to the British public as a national delicacy that isn’t revolting.

Oh, look, the Northerners heard that. They've stopped their internecine bickering and are once more on their way. If you're luckly, they'll stop for some fish and chips and give you a short respite.

Mentalism. Telepathy. ESP. Mind reading. Oh, speaking of ESP, when Sydney remarried they named their first child Edwin Sydney Piddington. ESP. Real class acts, those Piddingtons.

Again, I can't help but marvel here.

Mind reading. We’ve all wished we could do it at one time or another, whether to cheat at poker, manipulate people, or to save money on walkie-talkies.

Yeah, I used to think that would be a cool superpower. Then I started to realize what other people must actually be thinking, and now I'm grateful to just have my own twisted brain to worry about.

In fact, the Piddingtons were actually pretty respectable since they framed their shows as a puzzle for the public to solve rather than claiming they had godlike powers. We’ll get more into that later.

See? For this reason I can't hate the concept. If some people ended up believing the fiction, well, Barnum had something to say about that.

So what’s going on with the Piddingtons? To be honest, no one’s really exactly sure. Well, almost nobody.

And that's too bad; I know a lot of the allure of "magic" is a) trying to figure out how it's done and b) not figuring out how it's done. But after 70 years, you'd think their tricks would be as well-known as Houdini's.

There are lots of theories out there as to how the Piddingtons did it, and here’s mine: steganography. Not the more modern definition of hiding a file within the information of another file, but the more traditional meaning of encoding a message within mundane information.

Gotta side with the author here. Too bad he's about to get roasted by Northerners.

I’ll tell you what leaps out to me. Sydney Piddington has a stutter. He’ll trip over words and use lots of fillers – lots of “ums,” and “ahs,” things like that. It’s possible that’s just a vocal tic, but admittedly that’s kind of a weird one for a man who makes his entire living performing on stage and on the radio. There was an entire movie about one dude overcoming his stammer to give just one speech.

Throwing out a neutral reference to The King's Speech (an excellent movie, by the way) isn't going to save you. The British are coming.

Maybe that’s not how the Piddingtons did it (although I’m pretty sure I’m on to something). But the overall point is that there is always, always a trick with mentalists. And if they’re not admitting that there IS a trick, then it’s because they’re picking your pocket. While the Piddingtons are harmless fun, there is a more nefarious side to this.

Yes. Superpowers are just as often used for evil.

Because that’s the real tragedy about mentalists: the more unscrupulous among them prey on people who are looking for answers. The hopeless want nothing so much as hope, and they’re willing to pay big money for anyone who can give it to them.

It's one thing to knowingly spend money on entertainment for the sake of entertainment. It's another thing to be so gullible that you're going to be fleeced. It's the reason I sometimes gamble in Vegas, and also the reason I still have money after gambling in Vegas.

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