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 20 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 22 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.

February 24, 2020 at 12:12am

#976196

Tools

Language.

It's ubiquitous -- you're using it right now -- and yet its origins are as mysterious as that of life, the Universe, and everything. Of course, that doesn't stop us from speculating.

https://www.theatlantic.com/science/archive/2018/06/toolmaking-language-brain/56...

A Sneaky Theory of Where Language Came From
It might have hijacked our early ancestors’ brains.

I’ve come to see Bovaird, who teaches wilderness-survival skills in western Massachusetts, because I want to better understand the latest theories on the emergence of language—particularly a new body of research arguing that if not for our hominin ancestors’ hard-earned ability to produce complex tools, language as we know it might not have evolved at all.

I'm just going to throw out here the observation that when I think of "wilderness," one place that doesn't come to mind is western Massachusetts. But, okay. That's irrelevant to the discussion. I just thought it was funny.

It's well-known by now that humans aren't the only species to use tools, or for that matter communicate with each other. And yet, as far as I've been able to determine, we're the only extant species that uses tools to make tools. This nesting of concepts is, in my view, essential for language as we know it today.

Oren Kolodny, a biologist at Stanford University, puts the question in more scientific terms: “What kind of evolutionary pressures could have given rise to this really weird and surprising phenomenon that is so critical to the essence of being human?”

And here we start to steer into my least favorite subject: evolutionary psychology.

Stout found that his students’ white matter—or the neural connectivity in their brains—increased as they gained competence in flintknapping. His research suggests that producing complex tools spurred an increase in brain size and other aspects of hominin evolution, including—perhaps—the emergence of language.

Yeah, put like that, we're also steering perilously close to Lamarckian evolution

-- unless it's just badly phrased.

When hominins like Homo ergaster and Homo erectus taught their close relatives how to make complex tools, they worked their way into an ever more specialized cultural niche, with evolutionary advantage going to those individuals who were not only adept at making and using complex tools, but who were also able—at the same time—to communicate in more and more sophisticated ways.

Which puts us back on secure Darwinian footing.

Rudimentary language, which evolved in the context of toolmaking and teaching, was ultimately able to break away from its immediate contexts—this is the hijacking part—eventually employing those original cognitive pathways for its own unique purposes.

I think I see what they're getting at here. It's the development of the metaphorical aspect of language, an idea I find fascinating. Basically, at some point, we developed the capacity to relate something to something else that's superficially unrelated. For instance, you might count your sheep and find that there are 13 of them. And you might count the number of trees in your immediate vicinity and also get the number 13. The cognitive leap is to make 13 a separate concept - and suddenly, there can be 13 of almost anything that's countable. Or, if numbers somehow instill a feeling of deep dread, one might notice that the caudal appendage of a cat is similar to the caudal appendage of a dog, and rather than having separate words for the two anatomical protuberances, one might allow the word "tail" to do double duty - which can then be extended to describe what sticks out of the hind end of almost any animal that has one.

These are concepts we take for granted, but they represent massive cognitive leaps.

The article goes on to describe some pushback that this hypothesis has gotten, which is good -- though the article is clearly biased to the hypothesis. I'm not sure this particular article proves or disproves anything. As I always say when I'm not quite sure whether I like a beer or not, "further research is necessary."

Scientifically, the question is, okay, they've made the hypothesis of the connection between complex tool-making and language. What predictions does this make, and how can we test the predictions? Until more study is done, it remains just an idea. It's entirely possible, for instance, that other forces led to both the development of complex tools and the development of complex language; correlation doesn't necessarily prove causation, as is the case in every discipline.

Still, interesting enough for me to read and share the idea. After all, that's what language is for.

4 Like • Comment

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