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 2 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 4 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.

January 2, 2023 at 12:02am

#1042483

Born to Forget

My memory is like a steel sieve.

Forgetfulness: Why your mind going blank can be a benefit
We've all had those frustrating moments when we struggle to recall someone's name, or a key bit of information stays for too long on the tip of your tongue. It turns out these momentary lapses may actually be good for your memory.

I'm especially bad with names. This is one reason I don't get out much; people tend to think that I don't care enough to remember their names, but the fact is no matter how much I care, names just don't stick in my head.

But it happens to most people, to some degree, and with different things.

On 25 February 1988, at a performance in Worcester, Massachusetts, Bruce Springsteen forgot the opening lines to his all-time greatest hit, Born to Run.

One of many bits of trivia I do have etched into my brain is that Springsteen's birthday was September 23, 1949. That means he was 38 when he had his onstage brain fart. Hardly an age when forgetfulness becomes a problem for most people... unless it happens in public with a song you wrote 15 years (another bit of trivia seared into my skull) earlier and performed thousands of times since.

According to the conventional wisdom about the nature of forgetting, set down in the decades straddling the turn of 20th Century, this simply should not have happened. Forgetting seems like the inevitable consequence of entropy: where memory formation represents a sort of order in our brains that inevitably turns to disorder.

Funny thing about entropy: even memorizing something increases entropy.

Given enough time, cliffs crumble into the sea, new cars fall to pieces, blue jeans fade. As Springsteen put it in his song Atlantic City: "Everything dies, baby, that's a fact." Why should the information in our minds be any different?

1982.

That's when Atlantic City was released on the album Nebraska (look, music doesn't have to make geographical sense). I know I've linked the song in this blog before.

Want me to remember something? Make it a joke, or relate it to music. Especially Springsteen music. Yes, one of the reasons I saved this article was because of the Bruce hook.

In such a model, the preservation of information like song lyrics requires constant upkeep – which, in the case of Born to Run, no one could accuse Springsteen of neglecting. By 1988, he had certainly repeated the lyrics to his 1975 hit thousands of times over.

Yeah, sure, hit in '75, but work on the album started in '74 and that song, he wrote in '73 and early versions got played before the album came out. Hence, the 15 years I quoted above.

And so when he found himself staring out at the expectant Worcester audience at a loss, there was little he could do but confess into the microphone: "Sung it so damn much I forgot what the words were."

At this point, if you haven't looked it up already, you might be racking your brain to remember what the first line is, yourself. I'll save you the trouble: "In the day we sweat it out on the streets of a runaway American dream."

Anyway, there's a lot more to the article, though it doesn't lend itself well to cherry-picking quotes. Worth a read if you want to know where the current science of memory stands, especially how forgetting could actually be an integral part of memorization. Reading it, it seemed to me that the mind works analogously to a muscle: to strengthen a muscle, you don't just keep doing reps, but give it some downtime between weight-lifting sessions to heal.

The brain is, of course, not actually a muscle; I'm just saying the analogy struck me as useful.

One final note: the theory of spaced repetition mentioned in the article as a way to keep stuff in your head? That's apparently what Duolingo is based on. Despite my advanced age and decrepitude, a lot of the language learning actually sticks with me, but the road to memorization can be bumpy and twisty. I cannot forget the French word for seal (the animal), for instance, because I can make a pun about it in English, whereas the word for orchard, to cite just one example, often eludes me because I can't relate it to anything else.

It's still not clear to me how I'd be able to tell the difference between my usual forgetfulness and a Real Problem such as my parents had in their old age. Nor am I convinced that just exercising the brain to learn languages or solve puzzles can really stave off such problems. But that's no reason not to try; there are other benefits, not the least of which being bragging rights.

I'm just glad I'm not in a position to publicly flub my masterpiece.

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