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.

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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.

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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.

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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.

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June 16, 2024 at 10:11am

#1072725

Revisited: "A Whole Lotto Nonsense"

I try to do these retrospectives on Sundays, picking an old blog entry at random and checking in with the topic and how my thoughts may have changed. I have a few self-made rules, though, one of them being that any entry from less than one year ago is excluded.

Well, today's randomly selected older entry barely squeaks by that restriction, having been posted just over a year ago: "A Whole Lotto Nonsense"

The primary point of the linked article (which is still available) is to question the scary narratives trotted out every time a lottery jackpot gets big enough to attract the attention of reporters, or what passes for reporters these days. Apart from a few instances of lousy editing on my part, mostly involving punctuation, I don't think I'd change any of my commentary on it—not too surprising, as nothing major has happened in the past year to change my mind about that subject.

I also have another lottery story in the queue, but I have no idea when it might come up. Like lottery winners, I pick one using random numbers when I'm ready to do an entry. Could be tomorrow. Could hang out there for a year (in which case it probably won't go in this blog, because I have fewer than 300 entries left before it's full). So, spoiler alert: the particular article awaiting my public scrutiny will provide a somewhat more nuanced view of windfalls, showcasing some of the pros and cons, and focuses on just one data point (that is, one lottery winner).

Instead of rehashing stuff you can just click on a couple of links to read, today I'll provide an aside about probability and chance.

It's tempting to think that if there's a one in a million chance for something to happen, it will definitely happen if you try a million times, or that once it happens once, it'll take a million tries for it to happen again. You see this sometimes in reporting about floods and such: two "100 year" floods happen in back-to-back years, so some people scoff at the entire concept of 100 year floods (which, to be fair, hydrological statistics are kind of up in the air right now what with climate change, but their problem is generally with the math, not the science).

I'm not going to go into the math here; you can look it up elsewhere; I try not to bore or confuse people too much, and it can be confusing, even to me, and hydrology was something I actually trained in. But what a 100 year event really means is that it has a chance, based mostly on historical data, of 1% (1 in 100) to occur in any given year. The probability of an isolated event, such as a particular result of a die roll or a coin flip, is independent of any previous occurrences of the event. You flip a coin, it lands on "heads." You're tempted to bet that the next flip is "tails." This is the gambler's fallacy; in reality, the next flip has a 50/50 chance of being heads or tails.

Shuffle a deck of cards. Cut it. The chance of the cut card being an ace is 1 in 13. Shuffle again, and cut. What's the chance of it being an ace again? That's right; 1 in 13. On the flip side (pun intended), you could repeat this exercise 13 times and never once cut to an ace. No, I'm not going to calculate what those odds are.

Lottery odds are much smaller than any of these examples, but the idea remains. Part of the reason lotteries are described as a tax on people who are bad at math is because very large and very small numbers can be difficult concepts to deal with. When it comes to that, most of us are bad at math. Hell, it's hard for most of us to conceptualize the difference between 10 million and a billion. Yes, even me. It's not intuitive, like the difference between 1 and 100 is... and those are the same ratio.

All of which is to say: don't let the scare stories about lotteries affect your decision. There are plenty of good, actual reasons not to play the lottery (generally several million of them, as in your chance of winning is one in several million), but fear of bad shit happening to you shouldn't be one of them.

After all, unlike the numbers picked in the lottery drawing, you have some control over your attitude and actions (philosophical issues about free will notwithstanding). In other words, your chances of getting bitten by a shark drop to very close to zero if you never choose to go swimming.

And they were pretty damn low to begin with.

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