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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Write_Mikey_Write!

about 10 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 12 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.

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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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

February 5, 2024 at 10:48am

#1063550

The Streak

At first, I thought this was going to be another "you're doing it wrong" article, but no. Well, a qualified no. From Fast Company:

The Duolingo effect: How keeping the ‘streak’ is changing people’s behavior
Maintaining a streak is a major motivator, and apps have caught on.

"Duolingo" in the headline of course caught my eye, but that's not the only streak I have going.

An activity streak has the power to compel behavior, and marketers have taken note. Marketing researchers Jackie Silverman and Alixandra Barasch recently documented 101 unique instances, including Snapchat, Candy Crush Saga, Wordle, and the Duolingo language learning platform, of apps that have incorporated streaks into their architecture by tracking the number of consecutive days users complete a task.

Thing is, though, when I started with Duolingo, I was well aware that it was gamified. That is, it's deliberately structured to reward obsessive behavior, like how a fantasy role-playing game might dangle the promise of the next reward, or play triumphant music when you succeed at a quest, or whatever myriad little tricks game designers use to keep people playing.

That was, for a gamer like me, the whole point. Self-motivation almost never works for me; I get frustrated and go play video games.

But even that doesn't always work for me. The linked article displays someone's Wordle streak chart. Remember Wordle? I think it's still around. I got up to 101 consecutive days' wins, then failed one day. Rather than slog through trying to get another streak going, I just quit playing altogether.

Still, there's a difference (something the article does note later). Wordle doesn't have an overall goal, apart from the streak (I remember one day, my habitual first word, which was STEIN, was the actual answer, so there wasn't any motivation left to go for even that, anymore). Not so with Duolingo. My streak there is just the motivation; the goal is to become more facile at languages. Still French. At one point, I'd finished all their French lessons, but then they went and added more. A lot more. So I'm still slogging through French.

As an aside, some of the more recently added sections discuss cooking terms. You'd think that would be pretty damn basic in French, but no, many of those lessons were added fairly recently.

Based on input from people maintaining streaks and how streaks are described in the popular media, I suggest they have four underlying characteristics.

Keep in mind these are one person's definitions, not a generally-accepted scientific result.

First, streaks require unchanging performance and temporal parameters.

I'm skeptical about the "unchanging performance" thing; when I'm traveling, I have less time to maintain my streaks, so I let myself do the bare minimum; however, when I'm home, I do more.

Second, the streak-holder largely attributes completing the activity to his or her resolve.

I question this, as well. As I implied above, I consider my Duolingo streak to be mostly the result of the program's game-like design. This blog, in its fifth year of daily entries? Maybe that's my own stubbornness.

Third, a streak is a series of the same completed activity that the person maintaining the streak considers to be uninterrupted.

By Duolingo's own parameters, a participant can miss a day, occasionally, and have their streak automatically "frozen." This maintains the streak in their records through a day of inactivity. I would still consider it a streak if that happened, but in my 1622-day streak, that hasn't happened yet. The downside is that I find myself reluctant to go anywhere that doesn't have reliable internet access, but let's be real, here; that would be a consideration anyway, as my longest streak of all (besides being alive) involves being on the internet in some manner every day for 20 years.

Fourth, the streaker quantifies the series’ duration.

Which I just did for Duo. But I had to look it up. It's not like I spent all morning going "This is my 1622nd day for Duolingo." No, I just thought to myself, "Blog, then Duolingo lessons, and then I can get back to playing Starfield."

I have at least one other streak going, which is WDC newsletter editorship. I'd have to look up the numbers there, too, but I haven't missed an assigned issue of the Fantasy or Comedy newsletters since I started being a regular editor there. Both began in the late noughties. Those are monthly, though, not daily.

This definition distinguishes an activity streak from winning streaks and lucky streaks.

Wordle, as above, isn't necessarily an activity streak; unless you're really good, or you cheat, you're not going to hit every one. (If you really are that good, great! If you cheat, whatever; that's on you.)

There's a lot more to the article, but this is already getting long. Just a one more highlight:

Streaks can serve to gamify the underlying activity by creating rules and quantifying the outcome, and many people enjoy the challenge of a game.

And some don't, and that's okay, too. Everyone finds their own motivation. Or doesn't.

The author ends by suggesting starting a streak rather than making a New Year's resolution, though the date on it was January 3, by which point most people have already failed at least one resolution. By today, it's probably safe to go back to the gym. Which was a streak I had going for a few years, myself, until... you know what... made it hazardous to be around sweating, heavily breathing people.

That's a streak I might try to restart, but never in January.

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