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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10 Most Recent Blog Comments:

Write_Mikey_Write!

a day 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

a day 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.

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').

1 Like

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.

October 8, 2019 at 12:21am

#967431

Devolution

https://hedgehogreview.com/issues/animals-and-us/articles/the-deficient-animal

The Deficient Animal
Only the human species is capable of grasping, analyzing, and interpreting signs as symbols.

So, the idea is that the major thing that distinguishes us from other animals is... metaphor. The idea that something can stand for another thing is at the root of language, I reasoned, and we can also nest metaphors.

I've been saying this for years, but I don't have any credentials in the field, so mine is just another opinion. Still, I do enjoy some confirmation bias from time to time.

The article, of course, takes this a bit further. But the author engages in unnecessary obfuscation through esoteric language, so be warned.

The evolution from anthropoid to human was simply a function of the contingencies of adaptation and survival needs within different environments. This increasingly exclusive focus on biological similarities tended, on the one hand, to fold the human being entirely within the continuum of the animal order and, on the other hand, to minimize, downplay, or ignore altogether the distinguishing characteristics of the human species.

"Simply?" I don't think there's anything "simple" about evolution, except insofar as simplicity leads to complexity, but again - not really my wheelhouse.

This school emerged mostly from the work of German and Dutch biologists, zoologists, philosophers, and social theorists, including Paul Alsberg, Louis Bolk, Max Scheler, Adolf Portmann, Helmuth Plessner, Arnold Gehlen, Konrad Lorenz, and the British-American biologist Ashley Montagu, and has been appropriated and refined in recent decades by Jürgen Habermas, Axel Honneth, Hans Joas, Karl-Siegbert Rehberg, and others.

And the Name-Dropping Award of 2019 goes to... James Davison Hunter!

The consensus among the philosophical anthropologists was that the development of human beings was not simply a result of evolutionary progress, but rather of the inhibition of the evolutionary process.

Translation: we evolved by losing survival traits. As near as I can tell, anyway. Though that's not an "inhibition" but rather a new set of survival traits, I think.

In the late eighteenth century, Johann Gottfried Herder had called the human being “the deficient being”; others, following Herder, described humans as animals “not yet determined,” “unfinished,” “incomplete,” “physiologically premature,” and “organically deficient”—and, therefore, ever malleable. In addition to their unfinished character, humans also have no species-specific natural environment they can call home. Home can be anywhere and everywhere; indeed, humans are capable of adapting to a vast range of environments. They are, as the philosopher Max Scheler put it, “open to the world.”

That plopping sound you just heard was more names being dropped.

In sum, human beings must of necessity make up for their instinctual impoverishment by actively transforming the world to suit their own ends, mastering and re-creating nature rather than merely adapting to it. The means by which human beings do this is by representing the world symbolically, particularly through language, but more broadly through culture itself. Culture is a “second nature.”

And that's where this author goes beyond my simplistic assertion above.

The name (or sign) and the object to which the name attaches (the signified) are mediated by an interpretant. In other words, the meaning of a name or sign is manifested in the interpretation it generates in sign users.

Until, ultimately, you end up studying various academic subjects, leading to expressing basic concepts with four-dollar Latin-root words. This guy, by the way, is a professor at UVA, here in my hometown; and yes, they all write that way. (No, I don't know him personally.)

But we have to address the instances of symbol use in our close cousins:

Other species, most famously chimps, bonobos, gorillas, and other primates, are, in limited ways and experimental contexts, able to use signs as designators, combine words into simple sentences, use simple tools, and even create tools. They can do so in signifying, inferential, and ritual ways. They can also interpret the intentions of members of their own species—for example, intentions to mate, to hunt, to attack, or to eat. But they cannot use signs symbolically, as objects of reflection or analysis.

Only the human species is capable of grasping, analyzing, and interpreting signs as symbols. In other words, only humans are capable of using signs at a meta- or self-referential level.

It helps if said signs and symbols are limited to words of two syllables or fewer, but hey, you do you.

So, basically, if I'm understanding this right, the assertion is that without clear, innate survival advantages (claws, sharp teeth, speed, flight for escape, etc.) we had to have some other, more subtle survival advantage, and that advantage is our ability to transfer meaning from one symbol to another or to a referent. This is the foundation of both language and culture, and this adaptation is what eventually led to us doing shit like building skyscrapers and sending robots to explore Mars.

I may be projecting my own thoughts onto the essay, so if you got something else out of it (apart from a headache from all the professorese), feel free to chime in below.

1 Like • Comment

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