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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August 30, 2022 at 12:03am

#1037088

Complexity

Most of you probably know by now that there's a new merit badge available to the community, one I commissioned and, of course, The StoryMistress

implemented. Here it is, called "Complexity":

Merit Badge in Complexity
[Click For More Info]

Congratulations on your new merit badge! Thank you for supporting the Writing.Com community with your inspirations, participation and activities. We sincerely appreciate it! -SMs

While obviously named after this blog, it's not restricted; anyone on WDC can send it to someone else on WDC. But the meaning might be obscure to some; therefore, I will explain it. This may be the only time I ever do so.

Fair warning: math discussion ahead. But not a very technical one.

The header for this blog has always had a brief definition of what a complex number is in mathematics. As I've stated before, though, for me the title is a pun of sorts; both "complex" and "numbers" have other definitions. For example, the former can describe a kind of psychological disorder, and the latter also can be a synonym for a musical composition.

Also, all blog entries have an identifying number. Point is, I thought it was appropriate.

Now, there's a deceptively simple iteration you can do on any number, which I won't go into in detail (you can find all the detail you want by searching for it, or look at the Wiki link I provide below), only point out that when you do this iteration, the result either blows up to infinity, or it doesn't. If the number is a complex number, and the iteration doesn't tend to infinity, it's in the Mandelbrot set, usually graphically represented in black. Other colors are assigned to numbers outside the Mandelbrot set, depending on how quickly the iterations tend to infinity.

The cool thing, though, is that no matter how close you zoom in on a point on the boundary between "goes to infinity" and "doesn't go to infinity," you get the same sorts of spirals, whorls, and intricate designs. The boundary is self-similar at all scales, to any number of decimal places. (This is what I mean, in the blog header, by "Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.")

In the "real world," this can't happen. Zoom in closer and closer and eventually you get to atoms, and the stuff inside atoms, and there's a smallest size.

To me, this is a metaphor for how imagination can extend reality.

To see the image the MB is based on, go look at this article.

There's not a lot of math there, either; it's more of a philosophical essay and a discussion of the life of Benoit Mandelbrot, after whom the set is obviously named.

True to form, I'll quote from the article. (Just one passage, though)

Mandelbrot needed a word for his discovery — for this staggering new geometry with its dazzling shapes and its dazzling perturbations of the basic intuitions of the human mind, this elegy for order composed in the new mathematical language of chaos. One winter afternoon in his early fifties, leafing through his son’s Latin dictionary, he paused at fractus — the adjective from the verb frangere, “to break.” Having survived his own early life as a Jewish refugee in Europe by metabolizing languages — his native Lithuanian, then French when his family fled to France, then English as he began his life in science — he recognized immediately the word’s echoes in the English fracture and fraction, concepts that resonated with the nature of his jagged self-replicating geometries. Out of the dead language of classical science he sculpted the vocabulary of a new sensemaking model for the living world. The word fractal was born — binominal and bilingual, both adjective and noun, the same in English and in French — and all the universe was new.

Now if you want a more technical discussion and lots more pretty pictures (including an animation of a deep dive into the set boundary, showing its self-similarity), you can always peruse the Wikipedia page.

Unlike the boundary of the Mandelbrot set, this blog won't go on forever, and neither will I. This Merit Badge will, however, hopefully outlast both of us.

Oh, and if you want one, just comment below. I'm feeling magnanimous. That feeling won't last, so you have until midnight tonight, WDC time.

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