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

December 12, 2022 at 12:01am

#1041703

Colony

I don't have a lot to say about today's article, but it ties in with some other entries I've made; specifically, about the question of mind-body duality—or lack thereof.

The Gut Microbiome Helps Social Skills Develop in the Brain
New research in fish suggests that gut microbes can have a crucial early influence on the brain’s social development.

There are some things that are a bit problematic, but I'll get to that.

Two recent papers have shown that during a critical early period of brain development, the gut’s microbiome — the assortment of bacteria that grow within in[sic] it — helps to mold a brain system that’s important for social skills later in life.

One thing I've come to appreciate about Quanta: they don't usually bury the lede. The occasional typo, I can understand. Happens to all of us.

Scientists found this influence in fish, but molecular and neurological evidence plausibly suggests that some form of it could also occur in mammals, including humans.

So take it as a preliminary result, not settled science. If you corner someone at a holiday party and go, "You know, your gut microbiome has an effect on your social skills," a) you'd be jumping the gun and b) you'd better get your microbiome checked out, because your social skills suck.

In a paper published in early November in PLOS Biology, researchers found that zebra fish who grew up lacking a gut microbiome were far less social than their peers with colonized colons...

Snort.

...and the structure of their brains reflected the difference.

I mean, that's a big deal from what I can tell. I've suspected a connection between gut health and mental health for a long time (my mom had problems with both). Again, maybe or maybe not applicable to humans, but seeing a physical difference in the brain because of something in the intestines—something that's even a different genome—I can't help but think is an important finding.

In recent decades, scientists have come to understand that the gut and the brain have powerful mutual influences. Certain types of intestinal ulcers, for example, have been linked to worsening symptoms in people with Parkinson’s disease. And clinicians have long known that gastrointestinal disorders are more common in people who also have neurodevelopmental disorders, such as ADHD and autism spectrum disorder.

This is the problematic part. Autism isn't necessarily a "disorder," no matter what the DSM might have to say about it. But whatever the label, the important thing is the potential link between microbiome and behavior.

How these anatomically separate organs exert their effects, however, is far less clear.

Finding a correlation is one thing. Figuring out the mechanism is quite another, and it sounds like they haven't done much yet in that regard. There's some discussion of possible future studies later in the text.

Again, though, mostly I just saved this to link because it hints at some deeper truth about the physical and the mental, especially in terms of those microbes that are just as much a part of us as our own cells.

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