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.

1 Like

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

July 9, 2023 at 8:25am

#1052370

Revisited: "The Big Not So Empty"

I'm used to reaching back in time for these retrospective entries, but this one also reaches out into space: "The Big Not So Empty"

The link to the original article is broken, but by poking around on the Nautilus site, I found it

with a different URL.

This is good, because it's an exceptionally long articles, with fun and helpful cartoon illustrations, all focused on the goal of explaining to non-technical audiences what "space" is.

I have no need to rehash what I wrote back then, but one of the reasons I revisit these older entries is to see if anything's changed. And in this case, it has, sort of.

Near the end of the article, we have this:

And the exciting thing is that we are closer than ever to being able to probe these extreme deformations of space. Whereas before we were deaf to the ripples of gravitational waves moving through the universe, we now have the ability to listen in to the cosmic events that are shaking and disturbing the goo of space. Perhaps in the near future we will understand more about the exact nature of space and get at these deep questions that are literally all around us.

I remembered seeing something new about gravitational waves recently, so I went and looked for it, and, by cosmic coincidence, this was published just last week.

Now, I'm not going to do my usual commentary on that. I'm not averse to spouting off on shit I know little about, as you know, but in this case, this stuff is so far above my pay grade that I don't even know how to respond. I'm told that physicists are deliriously happy about these findings, though, and that link I just posted seems to do a pretty good job of explaining the significance. In brief, as I understand it, they found a way to detect incredibly tiny space-bending gravitational ripples from events in the early universe. Which is cool, and I think it's a nice complement to how the JWST is giving us better (photonic) images of stuff from the early universe.

I've even heard that they recently confirmed that time moved at a different rate back then, which... well, let's just say I need to make a run to the liquor store later to deal with all of this.

Just one last thing to note, which is more about the ambiguity of the English language: "space" in this context refers to, well, everything we know about, including, say, the space between things here on Earth, while "space" can also mean "the stuff outside the Earth's atmosphere." Outer space. Similarly, though more obscurely, you can sometimes see the term "gravity waves" in relation to weather phenomena

, but that has nothing to do with the gravitational waves these science articles speak of.

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