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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Show Recent Comments

August 22, 2022 at 12:01am

#1036797

Weight a Minute...

I don't have a lot to say about today's article; I just think it's a good, fairly simple, example of how science gets scienced.

Which Weighs More, a Pound of Stone or a Pound of Styrofoam?
It’s not a trick question: your brain answers differently, depending on whether the materials are part of the same object or not

Ah, but it is kind of a trick question, isn't it? There's objective weight, which can be measured by a scale, and subjective weight, which is what your muscles anticipate and feel. Consider two packages of the same weight, one large, one small (maybe the large one contains nothing but packing pillows). The small one will be easier to pick up and carry if only because of the bulk involved.

That's not what this article is about, though.

For more than a century, scientists thought they knew the answer to a curious question: why does 10 pounds of a low-density substance such as Styrofoam feel heavier than 10 pounds of stone? It isn’t heavier, of course, but repeated experiments have shown that it feels that way.

Science tip #1: Just because you think you know the answer, doesn't mean you do. Always test.

Now psychologists say their initial explanation may have been incomplete, and the new explanation could have far-reaching consequences, including for the way Netflix designs the algorithms that recommend movies to its customers.

Science tip #2: Your experiment doesn't have to have practical, everyday applications. But if you can come up with one, it'll be easier to communicate it to the teeming hordes.

The article goes on to describe the experiment and their results, which, as the headline hints, are that people are ass at guessing weights. Or something.

Knowing how the brain estimates weight isn’t just an interesting experiment—it can actually help scientists develop smarter technologies that we use every day. Now that we know more about how context changes the brain’s decisions, programmers might be able to update technologies such as Netflix to imitate the brain more accurately and provide more fine-tuned recommendations for users.

Article is from 2019. Netflix is still trying to recommend shows and movies to me that there's no way in hell I'd watch.

Anyway, like I said, not much else to say, and you'll have to go to the article to see what the actual experiment was, because it wouldn't be easy to take any of it out of context. And I'm all about easy. Give me the smaller package.

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