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

March 24, 2022 at 12:01am

#1029406

What Color is Your Cat?

As you know, sometimes I like to address the Big Questions in here.

Why do cats—and so many other animals—look like they’re wearing socks?
The possibilities of pigmentation are endless.

This is one of those times.

Grumpy Cat. Lil Bub. Maru. What do all of these internet-famous cats have in common? From ankle down, their paws are as white as the trendy marble countertops vying for attention in the very same Instagram feed.

I don't know about Instagram feeds, but I do have some experience with trends. Those marble countertops? One day, y'all will look back on those and they will seem just as outdated as Formica, or wooden wall paneling. All of those things that you laugh at now from the 70s and 80s? Yep, that's the fate of marble countertops, eventually.

The phenomenon of pigment mixed with white splotches can occur in pigs, deer, horses, dogs, guinea pigs, birds, and, in rare cases, humans. But it’s particularly prominent in cats, as evidenced by the fact that Socks consistently ranks in the top names for felines.

Me, I always wanted to name a pet Spot. Not just in honor of Data's cat from ST:TNG, but because when they start to annoy me I can open the door and go, "Out! Out, damned Spot! Out, I say!"

If the melanocytes are evenly distributed, the cat could have a unicolor coat, like Sabrina the Teenage Witch‘s all-black cat, Salem, or the all-white Hello Kitty.

Right, because the Evil Feline is totally a biological entity and not the product of some Japanese person's nightmare.

Piebaldism isn’t the only genetic quirk that can alter an animal’s fleece, according to the UC Davis Veterinary Genetics Lab. The tabby cat’s signature look is served up by the agouti gene, which determines the distribution of black pigment.

Another common name for a cat is "Tiger," because some cats have stripes, like tabbys. I remember reading somewhere once that back when there were a lot of tigers, before they were hunted to near-extinction by homo sapiens, they had the capacity to possess a calico coat, like the ones you find in domestic cats sometimes. I have no idea how true this is, but I find it hard to believe that it would have been conducive to survival.

Maybe the next meme-worthy cat should be named for a geneticist. Gregor Meowndel, anyone?

No.

One of my first forays into science was a high school project wherein I looked into cat genetics. Cat coat color isn't Meowndelian. Er, I mean, Mendelian. That is, sometimes you get strange results from breeding cats, as with calicos. But as it turns out, apparently, that's because the cat's pattern isn't strictly genetic, but also a matter of gene expression and conditions in the kitty womb. Kind of like with our fingerprints.

All very interesting, but you have to wonder what the point is, when it's well-known that whatever their fur patterns, they'll still knock things off of tables, beg for food, and try to fit into boxes.

Perhaps that last bit is because their genetic inheritance doesn't fit neatly into boxes.

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