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').

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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.

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

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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

September 4, 2021 at 12:01am

#1016728

Cryptonite

Just a few short hours ago, I had a few drinks with Sum1's Home

, who's visiting for a couple of days (and also with Artemismad Scientist

), so it's just as well that today's randomly-selected article is easy to deal with. It does, however, touch on a controversial and divisive topic, one that has been known to destroy friendships, end marriages, and divide siblings. Yes, I'm talking about... cryptocurrency.

Or, rather, Cracked is doing most of the talking.

5 Foolish Flaws That Make Crypto Total Trash

Well, I guess we know where the world's foremost intellectual site stands on the issue. Especially when they can't count.

Whether you think cryptocurrency is the second coming of Cyber-Jesus or that crypto is just stupid computer program money based on nothing (as opposed to fiat currency, which is also based on nothing), crypto's got some problems all the best Reddit-soaked minds haven't solved yet ...

Look, in the end, everything is based on nothing. The other alternative to fiat currency? Commodity-based currency. What's the actual, intrinsic value of, say, gold? Why, it's whatever people decide it is. Come the Apocalypse, gold won't be worth the paper it's printed on: it's shiny and pretty, but if you can't eat it or smite your enemies with it, it's worthless. What will be valuable after the complete collapse of civilization? Certainly not crypto, which requires electricity -- you know, that stuff that we won't have ANY of when the survivors emerge, starved and blinking, from their bunkers.

No, the Ruler of the Post-Apocalyptic Wasteland will be whoever has managed to stockpile the most coffee.

Anyway. Back to cryptocurrency, which I call Dunning-Krugerrands. As usual, Cracked counts it down.

6. Lose Your Password, Lose Your Money

One of the benefits of crypto is that it's decentralized. There's no central bank, no cabal of corporate overlords, Cultural Marxists, and whoever else we don't happen to like today, controlling your cash. Instead, crypto relies on a wallet, a password-protected application that holds your digital assets for you, like your own mini-bank (not of the spank variety).

Periodically, I forget the password to my primary bank account. I can recover it via a link (assuming I can remember the password to my email account). I don't think it works that way for Dunning-Krugerrands.

It's true that you can call up a professional to use brute force attacks on a crypto wallet to try and unlock it if you want to put your economic future in the hands of Dave Bitcoin.

So... it's not secure?

5. Letting Someone Else Hold Your Bitcoins ... Doesn't Always Go Well Either

I can't find a decent excerpt here. You're just going to have to go to the link to see how bitcoin exchanges can suck ass.

4. Crypto Anonymity is Only For the Sophisticated

Police are actually getting pretty good at tracing and seizing the stuff.

So... yeah, this. Think you're safe from prosecution because you hired that hitman (who was really an FBI agent) with Dogecoin? Such naïvété. wow. much laughter.

3. Crypto is Actually Hard to Use

Crypto relies on digital infrastructure alone, which means that if that infrastructure screws up, you can't buy a car, pay for pizza, or the bootleg Ninja Turtles boxset you've had your eye on. You're totally dependent on it to transact business.

Again, utterly useless come the Apocalypse. But even if we manage to escape that, how many times a day does YOUR internet glitch?

2. All Our New Renewable Energy Capacity is Being Eaten by Crypto

Hell, for that kind of energy, you could just go mine gold. Like, out of the ground and everything.

This has been bugging me for a while. So much energy is needed to mine twitcoin now that, well, there have got to be more efficient ways to make a profit, right? At some point, it will become cheaper to invent a fusion reactor that can literally turn lead into gold... at which point, gold will lose what value it has left.

1. Inequality is a Huge Problem in Crypto Too

Another one I can't do justice to. This isn't about racial inequality, though, but the ability for large players to do market manipulation.

You know, just like with fiat currency.

I'm not trying to come down hard against cryptocurrencies, mind you. I don't know enough about them, and I know I don't know enough about them. I just think it's important to know potential downsides whenever trying anything new. If you can still live with the disadvantages, great, have at it. But I've encountered too many people (all online, naturally) who have absolutely converted to the cryptocult, and there's no swaying them.

And it's still a sad state of affairs when those disadvantages are most clearly enunciated by a comedy website.

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