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.

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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.

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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

April 23, 2023 at 8:59am

#1048643

Revisited: "Cryptonite"

Today is the day for me to delve into the past to see what I was up to then and how things might have changed.

The RNG took me all the way back to September of 2021 this time, so let's see what I had to say in "Cryptonite"

...

...a controversial and divisive topic, one that has been known to destroy friendships, end marriages, and divide siblings. Yes, I'm talking about... cryptocurrency.

So not much has changed.

Unsurprisingly, the original article link,

from Cracked, is still there.

It's almost inevitable that I've picked up a few things in the year and a half or so since I wrote the entry, but I still won't touch crypto, not even with someone else's pole.

If I remember the timeline correctly (and I probably don't), it was the following February that crypto was pushed heavily during the big sportsball game commercials. Not that I saw them, but it's hard to avoid hearing people talk about them. After that, the price of the leading cryptocurrency, bitcoin, crashed.

So, not trusting my own memory, I looked up this chart,

which tracks btc vs usd for the last 2 years.

Such are the perils of hindsight (and my memory). Someone could easily have come up to me a month after I wrote that September entry and crowed, "You're an idiot. The value of bitcoin has skyrocketed!" Because it had. And then I could have come back pretty much any time after that and said, "No, you're the idiot," because it never again reached the stratospheric level of October 2021 prices. Or even September.

Thing is, though, that whatever the value of it, the fact that it's quoted in US dollars doesn't say much about the intrinsic value of Bitcoin, but it says a lot about the intrinsic value of the USD.

It's basically a gamble. Not that I think there's anything inherently wrong with gambling; I do it, myself—on an occasional basis, and always with a self-imposed limit that is so low that it's basically my entertainment budget. Other people go to sporting events or rock concerts or Broadway plays; I sometimes go to casinos. So it would be hypocritical of me to dismiss it just because it's gambling.

"But the stock market is also a gamble, Waltz!"

Well, no. I mean, it can be, if you focus on short-term trading, which is what I think most people do because you're always hearing about day traders and the Robinhood app or whatever. But there's a big difference between investing—that is, long-term buy and hold, much less sexy than the adrenaline rush of day trading—and gambling.

A stock generally has some intrinsic value (which is almost always lower than the share price). This is because it represents partial ownership in a company, which will ideally have assets and a semi-reliable income stream, putting a floor on its valuation. Sure, sometimes individual companies get caught cooking the books (Enron, e.g.) or their income stream dries up, or they take on more debt than they have assets, and the share price falls to near zero. But more often, they have an interest in keeping on making money, which drives up the share price because people think it'll be more valuable in the future (this is a gross oversimplification, I know). Overall, the latter outnumber the former, which is why investment advisors harp on about diversification: the wins tend to more than make up for the losses.

There's still risk. But the analogy I like to make is that speculation and short-term trading are like playing in a casino, while long-term investing is like owning the casino. You can still fail, but you have the house edge.

But what's the actual inherent value of crypto? From what I can tell, it's pure speculation.

Still, no, it's not the gambling aspect that keeps me away. It's the same reason I shy away from gold trading: advertising. Also, to a lesser degree, not knowing the odds.

It's not just my inherent hatred of ads, either. Consider: if someone pays a premium for a 30-second spot during Big Sportsball Game, that means that they: a) want your fiat currency more than they want whatever they're selling; and b) think that they can make enough fiat currency to offset the high cost of said commercial. Now, to be fair, sometimes what they're promoting isn't the crypto itself, but a new way to store it. Like banks, which run commercials all the time. Since I wrote that original article, at least some of those crypto exchanges have utterly failed, the most public of which was the very well-covered fall of FTX. Still, even in that case, point b above applies.

So if someone is promoting, say, Dogecoin on TV, they're already giving away their game, which is "I'd rather hold dollars than dogecoins." If dollars are more valuable to them, taking into account the cost of advertising, why shouldn't they be more valuable to you?

I'll never ding someone just for wanting to gamble, because, like I said, I do it myself. Though I have an awareness of the odds, and I don't treat it as a get-rich-quick scheme.

But hey, there's at least one actual, legitimate, and valuable use of Bitcoin: Last time I checked, which was around December, WDC is taking it as payment for membership.

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