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.

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 26, 2022 at 12:01am

#1038155

Cloverleaf

Speaking of luck, my random number generator popped up this article right after yesterday's piece about Fortune.

The Secret to Being Lucky
Everything happens for no reason.

This one's from Wired, so it's probably an easier read than yesterday's.

Really, what more can or should be said about the future? Look around and see what happens. You can look for your crypto windfall. You can look for the love of your life. You can look for the queen of hearts. Seek and ye might find. You can even look for a four-leaf clover, though the chances are about 1 in 10,000. But if you find one, the shamrock is no less lucky because you looked for it. In fact, it’s luck itself.

Sounds more "looky" than "lucky." Also, a shamrock is a particular clover-adjacent plant; the four-leaf clover can be pretty much any clover. Except, probably, alfalfa. Which may or may not be considered a variety of clover. It's all very complicated and I can't be arsed to sort it all out tonight.

“Diligence is the mother of good luck” and “The harder I work the luckier I get”—these brisk aphorisms get pinned on Ben Franklin and Thomas Jefferson, lest we earnest Americans forget that salvation comes only to individuals who work themselves to dust.

Hahahaha.

In truth, the luck = work axiom does nothing but serve the regime and the bosses, by kindling credulity in a phantom meritocracy instead of admitting that virtually every single advantage we get in the world is one we lucked into...

Which is what I've been saying.

How about we invert the meritocratic fallacy in those aphorisms and create a new aphorism that makes “work” the delusion and “luck” the reality? “The luckier I get, the harder I pretend I’ve worked.” An excellent way to describe the people born on third who believe they hit a triple.

There you go. That's more like it.

Disclaimer: I'm lucky. Fortunate. Whatever you want to call it. Life is fundamentally unfair, but it's usually unfair in my favor. I make no apologies for this, nor seek penance. But I do consider it my obligation to point out that luck is a far more reliable indicator of success than hard work is. As I've said numerous times, if hard work were all it took to achieve financial success, sharecroppers would be billionaires.

After all, the chances of the precise sperm colliding with the exact egg in the right fallopian tube and convening to make you—or me—are so low as to be undetectable with human mathematics.

Yeah, no, it's detectable, though very low. It's also irrelevant. As I've also said numerous times, no matter what the odds were, once you've won the lottery, the odds of having won the lottery are 1:1. All the "you were extremely unlikely" phrase accomplishes is putting an upper limit on how many other people might share your precise genetic makeup.

But work and diligence can never be the parents of luck, because luck has no mother, no father, no precedent or context. Luck is a spontaneous mutation, signaling improbability; it shows up randomly, hangs around according to whim, and—as every gambler knows—makes an Irish goodbye. Mischievous luck is fun, a shamrock, a “lady.” It’s worlds away from grinding toil.

"Irish goodbye" is probably more than a little on the stereotyping side, having associations with other negative Irish stereotypes. But whatever; I didn't write it.

The "spontaneous mutation" bit reminded me of a story by Larry Niven. In Ringworld, one of the characters is lucky. Turns out that, in the story, she was the result of several generations of people winning the birth lottery (or some such; it's been a while). Point is, he treated luck as a genetically inheritable trait, rather than a matter of circumstance.

Einstein didn’t like the idea of God “playing dice” with the world. Lucky for Einstein, dice, in a world determined by luck, are not thrown by anyone, much less a God who is said to have Yahtzee skills. Instead, the chips fall where they may—and really they just fall, unpredictably, spontaneously. We then look for patterns in them.

Einstein's comment is often misrepresented, so I'll point out two things: 1) He apparently didn't believe in any of the traditional interpretations of God, instead using that word as a three-letter shorthand for "the laws of physics." 2) That particular quote was his visceral reaction to the idea that, at base, everything is probabilistic, as implied by quantum physics. Thing is, he was a pretty smart dude, and it's my understanding that evidence brought him around to randomness in the end.

Living by a doctrine of luck promotes at least five excellent things that have got to be good for your brain.

"have got to be" is an opinion, but let's look at the "five excellent things" and form our own opinions.

1. Active skepticism about “meritocracy.”

Yeah, okay. I've railed on that in here before, and probably will again.

2. Recognition of the utter contingency of one’s own advantages. An act, if I may, of “checking your privilege.”

Again, I recognize it; I check it; and I wallow in it.

3. Appreciation for the spontaneity, serendipity, and unpredictability of the universe.

I don't know if that's good for one's brain or not, but it does bring me joy.

4. A way to practice “gratitude” without doing calligraphy in $75 journals. All you have you do is say, every time it hits you that life is OK and could be otherwise, “What luck!”

Ugh.

5. A way to make more luck in your life.

Which is the point of the article (if the article can be said to have a point at all), and I'm still not fully convinced.

Anyway, like I said, it's sheer luck that this came up back-to-back with yesterday's; I have a few dozen articles in my queue and not many of them deal with this subject. So you probably won't hear more on this from me for a while.

Lucky you.

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