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:

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

March 10, 2023 at 12:12pm

#1046182

Musk I?

Another one from "JAFBG"

[XGC] today.

Choose one famous person and rant or rave - your choice.

Funny thing about "rave." It's not precisely a contronym, like "cleave" or "sanction." But it can mean "talk crazy talk" and "heap praise upon." Unlike with "cleave," it's possible to do both. In fact, any raving I do about someone famous would probably be both.

And praising someone presents risks. I seem to remember, several years ago, praising Joss Whedon as a director. Well. That didn't age well, did it? Once it was revealed he was behaving like a douchecanoe, no amount of good that he did could ever make up for that. No matter how great someone is, at some point, they'll turn out to be human after all, and you'll look like an idiot for ever supporting them, because people are unforgiving. So praising someone has its dangers.

On the contrary, once someone reveals themself to be a cockmuffin, no amount of good stuff that they do can ever make up for that. They are always, forever, known as a cockmuffin, because, as I noted, people are unforgiving. That's why it's easier to rant about someone than to praise them: by ranting, you will always be right; if you rave, you'll eventually be laughed at for being so hopelessly naïve.

The risk, there, even with public figures, is that there are laws about talking too much shit about someone. In the US, from what I understand, it has to be both shitty and untrue. But I don't understand much, so I try not to shit-talk a lot.

But Muskmelon is too rich a target not to.

Oh, I admit, I was, like many people, taken in at first by his apparent business sense. I loved the stuff SpaceX was doing, and cheered the first time I saw a rocket booster land itself. I thought Tesla was doing great work: introducing electric cars that didn't look or handle like you were doing penance. Good stuff, all of it. Top of the world.

And then the raving began.

I don't remember all the details. Something about a bunch of people trapped in a cave, and false accusations of pedophilia? Making promises he couldn't keep, like with tunneling machines or high-speed rail? Some half-baked (actually, fully baked and burnt) scheme for founding a colony on Mars? Managing to turn public opinion against autonomous vehicles by leaping them into production before they were ready? I don't know. The details don't matter.

Taking over Birder and sinking it into the lava pits of Hell was just the cherry on top of the parfait of doom. I never liked that site anyway, so it barely affected me, but I still felt like it made Rachel Carson's prophecy of "Silent Spring" come true.

I get the feeling that if he hadn't had those initial successes, Muskmelon would be just another cracked pot. But he did, which makes the crash-landing even worse.

Maybe he should have paid more attention to his engineers. Not just the ones at the bird site, but the ones at SpaceX, who became really good at soft, controlled landings. Instead, he'll now be forever known as a lithobraking expert.

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