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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July 12, 2022 at 12:02am

#1035065

Imagine

It's not very often that I talk about the plain meaning of this blog's title. I generally let the various puns on it rule here. "Complex" has several meanings, as does "numbers."

But today is one of those days. I know a lot of readers don't "get" math, but the video I'm linking today is more about history than it is about equations and such—though there are certainly equations used to illustrate the points. Periodically (pun intended), I go down a YouTube math/science video rabbit hole (though I have enough sense not to link a lot of them here), but I can't remember if this link was the result of one such journey, or if someone brought it to my attention.

Since it's a YouTube video, I'm linking it here as usual, but also embedding the video if you can't be arsed to click through.

How Imaginary Numbers Were Invented

Since it's a video, I can't do my usual quote-mining here, but to paraphrase what I feel are the important takeaways: Mathematics started out as a way to quantify the world. As such, things like negative numbers were anathema for a very long time (you can have three oranges, but you can't have negative three oranges). It was only when mathematicians let numbers be the abstractions that they actually are that math became a powerful tool for science, paradoxically allowing us to gain a much deeper understanding of reality.

Such abstractions shouldn't be so difficult to understand. As writers, we work in abstractions too—we call them metaphors, and as with math, they often reflect a deeper reality than the one in front of our noses.

This, by the way—the ability to think in metaphor—is, in my estimation, the thing that makes humans different from all other animals on Earth. It's why we're able to send rockets into space, some of them containing powerful space telescopes

that search deep into space and time, giving us an even greater understanding of the universe.

(The link in the previous paragraph is to the BBC article talking about the first image released from the Webb telescope, just yesterday. I have some issues with the way the article describes things, but this entry isn't about that, and it's good enough to get a sense of what's going on. Also the picture is really damn awesome.)

So it turns out that so-called imaginary numbers and, by extension, complex numbers, really do describe reality at its most fundamental (so far as we've been able to determine) level.

A far cry from counting grains of wheat, for sure. But it all comes from the same mindspace.

The video quotes Freeman Dyson (yes, the guy who conceptualized the Dyson sphere): "...nature works with complex numbers and not with real numbers."

The phrase "real numbers" has a specific definition in mathematics, but the rest of us have a different conception of the meaning of "real." Using the layman's understanding of "real," well, all numbers are real, even the imaginary ones that don't seem, at first glance, to correspond to anything we can see or touch.

Or, alternatively, all numbers are metaphors, describing an abstraction of what we can see or touch.

Either way, math is just crazy good at describing the world we know of as "real." In many cases, such as the one in the video above, the math existed first, only to later find a use in science.

It's all real. Even the stuff we don't understand. Especially the stuff we don't understand.

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