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

October 21, 2020 at 12:01am

#996383

Living the Dream

I don't have a lot to say about today's link, but I'll say what I do have. Here it is:

Is Dreaming Real?
When you’re lucid, it can feel so real the distinction ceases to matter.

As you've probably noticed, I'm not one to resort to dictionary definitions of things. Dictionaries are descriptive, not prescriptive, and the definitions often have a subjective element.

That said, let me put up the pertinent Oxford Dictionary definition of "reality:"

the state or quality of having existence or substance.

And the stated philosophical definition:

existence that is absolute, self-sufficient, or objective, and not subject to human decisions or conventions.

I find these definitions problematic. "having existence or substance" just kicks the can down the road, forcing one to try to define "existence" and "substance." A good example is the question posed in the title of the article above: A dream has existence, if not substance, but that "or" in there means that either condition can be met. A photon has existence, but no substance. On the other hand, I can't think of anything with substance that doesn't possess existence - that would be a chunk of matter that doesn't exist. Perhaps you can think of one.

As for the philosophical definition, perhaps I'm being dense here, but our decisions create things that are objectively real on a regular basis. I can decide to bake a cake, and an hour later, behold, there is a cake.

It could be argued that there's a Platonic ideal of "cake," that it, in a sense, has always existed, will always exist, in the realm of the possible. I promise you, any cake I bake will not be ideal, Platonic or otherwise.

But the realm of Platonic ideals is itself a thing that has no physical, substantive existence. Plato thought that shit up. It's a mental construct. Oh, I suppose it could correlate to a particular arrangement of neural firings in his brain and that of his students, on down to the present day and hopefully beyond. That's hardly what we think of when we consider "reality."

Being a pragmatist of sorts, I tend to my own, ideosyncratic definition of "reality:" that which is still there when we don't believe in it.

I'm not a philosopher; I just read about them. So I'm sure my own definition leads to all kinds of roadblocks and paradoxes, too. Consider it a working definition, a practical tool to help me divide the real from the unreal. The chair I'm sitting in is real and won't disappear if I stop believing in it. It won't even disappear when I die; someone will have to decide what to do with the damn thing. I can imagine a much better chair, one that's more comfortable and with hydraulics that won't jar my spine by giving out every time I lean back to stretch. If I stop believing in this ideal notion of "chair," or die, then poof, this ideal chair exists nowhere. Well. Unless you count this paragraph.

Point is, any attempt to define what is reality and what isn't is a bit like gripping a bar of soap, or trying to nail Jell-O to the wall.

Getting back to dreams, though, as is so often the case on the internet, the headline is somewhat misleading. It's not asking these deeper, subjective, and possibly unanswerable questions; it mostly talks about lucid dreaming, and the difference between being a passive observer in one's dreams and taking an active role in their unfolding. There are implications on dreams' effects on our minds, and hence our brains -- which, for most people who aren't politicians, I'm prepared to postulate the real existence of.

Consequently, the article is interesting -- which is why I linked it in the first place -- but I feel like it asks the wrong question, precisely because we can't really define what "real" is. We all dream, whether we end up remembering them or not. Since it's a subjective experience, a dream unremembered might as well never have happened at all. Most of us, myself included, only remember fragments of dreams, with a vague idea that more happened but we just can't seem to grasp what it was.

I don't doubt that, as the article suggests, unlike Vegas, what happens in dreams doesn't always stay in dreams. It can impact our thoughts in the waking world. This can even change other peoples' entire perception of the world; the article uses Einstein's dream as an example, something that led to one of the most important scientific breakthroughs in history and has had an effect on all of us, whether we believe in it or not.

And if something changes perceptions or actions in consensus reality, is that something not, in some sense, real?

Well. Like I said. I don't have much to say about the article itself, but it turns out I had quite a bit to say about its philosophical underpinnings. There are those who assert that we each create our own reality, and there are certain interpretations of quantum mechanics that seem to bear that out -- though by no means all of them.

Perhaps those who remember their dreams can do so more effectively than others.

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