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 17, 2022 at 12:02am

#1041865

Facts Machine

You know, it would help if some of these were backed up by... you know... science. Or at least links.

15 Incredible Science Facts That Will Blow Your Mind

16. The mind can't actually be "blown" by information alone.

17. There are links to support that site, all of which take crypto. Consequently, I don't trust a goddamn thing it says. Any resemblance to "facts" is coincidental.

18. I browse using an ad blocker and a script blocker. I have no idea what the site looks like without these things. Click at your own risk.

Now, the actual "article."

1. The human brain receives and stores 11 million pieces of information per second but is only consciously aware of 40.

Not even getting into the definition of "piece of information," the brain doesn't "store" shit. We're not computers. That may be the most popular metaphor for brains right now, but that doesn't mean it's anything but a metaphor.

2. In the time it takes to read this sentence, approximately 1 billion neutrinos from the sun passed through your body.

I'm not going to look that up. It seems about right. Thing about neutrinos is they don't interact with ordinary matter very much (which is why they're bloody damn difficult to detect), so this is a great big "so what."

3. Something to remember next time you have a cold, a handshake transfers more germs than a kiss.

While there may be some truth to this,

I generally don't shove my hand into my mouth after shaking hands, and skin is a pretty effective barrier to most "germs."

This is sensationalist bullshit on the same lines as "your phone is dirtier than a toilet seat." Ooh, something that doesn't get sanitized on a regular basis is dirtier than something that (theoretically) does: film at 11.

4. Russia has a bigger surface area than the former planet Pluto.

Ugh, this shit again. Pluto never changed; only its classification has. You know how people out west look at the Blue Ridge Mountains and scoff, "Those are just hills!"? Yeah, it's like that.

5. If you were able to drive your car at an average speed vertically, you would reach outer space in about an hour.

All this is doing is restating the commonly agreed-upon boundary between "atmosphere" and "space," which is 100 kilometers above sea level. This too is misleading because there's no hard boundary; air just kind of gets thinner and thinner. So it's assuming that 100 kph is an "average" speed, which may be true out in the boonies, but it's still misleading.

6. Sharks can be long lived – up to 100 years.

And tortoises can live even longer. Interesting but hardly "mind-blowing."

7. The oldest known living creature, a clam called Ming, was 507 years old when it was accidentally killed by the scientists studying it. Its age means that it was born when Henry VIII was on the throne of England.

There are older plants. Also, an argument can be made that since some amoebae reproduce by fission, and there's no way to tell which was the original amoeba and which is the clone, every such amoeba alive today has been alive since amoebae first evolved.

8. There is a method by which peanut butted can be turned into diamonds.

I'll ignore the typo, but yeah, duh, it's chock full of carbon. Anything organic (chemical definition, not just nutritional) can be turned into diamonds. There was (maybe still is) a company that turned cremated peoples' ashes into diamond.

9. Eating ice cream can give you “sphenopalatine ganglioneuralgia” – better known as brain freeze.

Which would be the exact opposite of blowing one's mind.

10. Plants can recognise their relatives and work alongside them to grow stronger.

Newsflash: evolution favors cooperation. Yeah, I know, the "competition" aspect is what ordinarily gets played up. But we wouldn't have taken over the world without cooperation.

11. On Venus it doesn’t just rain metal, it rains two different types of metal – galena and bismuthinite.

I mean, okay. Those are minerals, not metals. The former contains lead and the latter, bismuth (shocking, I know), which are metals. And to an astronomer, anything heavier than helium is a metal. So... technically? It's also hardly news that Venus' atmosphere is hot enough to melt certain metals.

It also "rains" sulfur dioxide, apparently.

12. According to science, the chicken came first – only later did the species evolve the egg as a means of bearing young.

Okay, this has got to be a "gotcha" item. It's so wrong it's hard to figure out where to start with how utterly, stupendously, and completely wrong it is. I'll just note that eggs have been a thing since the beginning of sexual reproduction, and I can't be arsed to look up exactly when that was but a billion years sounds like it might be about right. Chickens, and for that matter other birds, are relatively new to evolution. At some point a proto-chicken laid an egg, and boom/crack, out came a chicken (evolution doesn't actually work that way but let's leave that for now). So there's no doubt in my mind that the egg came first.

In the interest of fairness, though, I will post {xlink:https://www.newscientist.com/question/came-first-chicken-egg/}this, which makes what I consider a convoluted argument that the chicken came before the egg.

Feel free to post that old comic with a chicken and an egg in bed together with the "I guess that solves that age-old question" caption, and I'll feel free to ignore it.

13. Up to 65% of autistic people are left handed.

"Up to" is weaselly as fuck (it could mean anywhere from 0 to 65), and it's reprehensible to imply that there's a causal link. (There is evidence that left-handedness is more common in autistic people than in the neurotypical, but I doubt the 65% number.)

14. Scallops have up to 100 eyes – usually blue in colour.

There's that "up to" again. It's wrong, anyway; the actual "up to" number is 200.

15. As you read this, you and the earth are spinning at 1000 miles per hour and moving through space at 67,000 miles an hour.

That first number depends on where you're reading this. Are you on the equator? Then yes. Actually a bit more than that. Are you in Antarctica? Then you're hardly moving at all, and not just because you're busy freezing to death, but because anything closer to the axis moves slower (angular velocity is the same, but that's not what miles per hour measures). At my latitude, it's about 820 mph.

The second number seems about right for Earth's orbital speed around the sun.

The difficulty is that the sun is moving too, orbiting the galaxy. The linear velocity there is something like 500,000 mph. And the galaxy is moving. You can't say how fast you're going unless you also specify what that's relative to. Here, I can't be arsed

to explain it better.

Bottom line: don't trust "factoids," especially when they come from sites funded by Dunning-Krugerrands.

Don't trust me, either, of course, but at least I did the tiniest amount of research and I'm not asking for donations.

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