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.

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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.

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.

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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.

2 Like

November 14, 2020 at 12:07am

#998322

Epiphany?

It seems that the word "epiphany" originally had the connotation of divine insight.

PROMPT November 14th

Describe an epiphany or "a moment when you suddenly feel that you understand, or suddenly become conscious of, something that is very important to you" that had a profound effect on your life and/or personal opinions.

It's been said that Newton's theory of gravity finally came together when he got beaned by a falling apple.

This is almost certainly apocryphal, like Washington and the cherry tree, but the persistence of the myth speaks to something in our collective psyche.

Apples seem to hold a special place in our collective metaphorical milieu. Perhaps most famously, the fruit of the Tree of Knowledge in Eden is popularly portrayed as an apple (though the original Hebrew text seems to contradict that as well). In the story -- and for the sake of this discussion I'm going to present this from my point of view, which is that the whole thing is allegorical -- the fruit, apple or otherwise, represents the moment when humanity developed self-consciousness, as illustrated by the immediate invention of proto-clothing. In short, it is a metaphor for whatever spark it was that guided us along the path to understanding the universe and our place in it.

In my view, this was the best thing that ever happened to humanity -- not a fall at all, but an ascent.

Which is why the original connotation of "epiphany" is amusing to me; that original Ascent was purportedly the exact opposite of what God wanted, except that if God was all-knowing, then God had that shit planned all along. Original Sin? More like Original Entrapment. Still, according to the story, the insight wasn't divine in origin.

Anyway, it should be no surprise that the popular myth about Newton's epiphany survives. It was Knowledge itself that hit old Isaac upside the head.

I'm no Newton, so my own epiphanies are far more mundane. So mundane, in fact that, the only ones I can think of right now are a) one concerning emotion that I know I've talked about in here before and b) my realization that the Earth, and everything on it including us, is made of star material -- which was honestly mind-blowing when I first realized it, but nowadays it's been mentioned so much that it would be trite if it weren't so intrinsically awesome.

It's not strictly true, of course. A significant number of atoms in the human body are hydrogen atoms, which could well have been primordial; that is, left over from the Horrendous Space Kablooie (as Calvin called the Big Bang). Pretty much everything else, though? A product of nuclear fusion, which these days happens in stars.

But there's one thing about it that still eludes me, and perhaps I'll learn about it someday or maybe have an epiphany; I don't know. I'll see if I can explain it in a way that makes sense. I call it the Neutron Problem.

In the early universe, once it got to the point when what we call elementary particles started to form, the dominant element was hydrogen. Hydrogen consists of a proton and an electron (though it took a while for the electrons to be able to bind to the protons). There's an isotope, deuterium, that adds a neutron to the mix. In this primordial era, some of the deuterium fused to create helium (and apparently some lithium too, but insignificant to what I'm talking about).

So what you end up with is a bunch of hydrogen, including some deuterium; and a fair bit of helium. The deuterium and helium contain all of the neutrons. All this gas clumped together and eventually made stars which proceeded to do their own nuclear fusion, eventually creating elements with more protons, like the carbon, oxygen and nitrogen that we need to live.

But here's the thing: free neutrons decay quickly. Like, half-life of 15 minutes quickly. (Protons, in contrast, don't decay, or so I'm told. Nor do neutrons bound in nuclei. There might be exceptions but they're irrelevant here.) So it's not like there can be a lot of free neutrons buzzing around. And yet, nuclear fusion goes on in stars and supernovae, and it doesn't take long before it starts making elements that have more neutrons than protons in their nuclei.

So what I wonder is where the excess neutrons come from in order for fusion to create heavier elements.

Now, this might very well be so basic that I missed it in high school chemistry and physics. Or I'm just not looking in the right places. I'm certain the answer is out there. But when I google something like "where do neutrons come from," the answer is always like "nuclear fission and radioactive decay of heavy elements," which doesn't tell me where the damn buggers came from to make the uranium or whatever in the first place.

I know that when a big enough star collapses, it makes what's called a neutron star because of... well, honestly, I don't remember all of the science, but basically the electrons and protons smush together under the intense gravity and make neutrons, so you end up with a gigantic dense ball of neutrons. But it's not like the little bastards can escape the gravity field, and even if they did, again, 15 minute half-life unless they happen to find a proton to marry.

So anyway, once I hear the answer to that it'll either be like an epiphany, or I'm going to facepalm and go "I should have realized that."

But I didn't, because I'm no Newton.

Maybe if I ate more apples...

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