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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 11 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

July 1, 2024 at 10:03am

#1073404

Pedantonyms

Today, from Cracked, we have an article that should be of interest to writers.

5 Evil Mirror Versions of Words We All Use
You’ve heard about sparkles. Now prepare to darkle

Yeah, people avoid me because of my darkly personality.

Words have opposites. That’s one of the first thing we learn about words, thanks to Grover demonstrating to us the difference between “near” and “far.”

The truly messed-up thing is, I remember that bit. I can't remember what I had for breakfast yesterday, but I remember Grover's running and shouting with near-perfect clarity.

Well. Probably not. Memories change. But at least I know what he's talking about (in the words of Captain America: "I understood that reference!")

Some words, however, have no opposites. Think about “lie,” for example. We’re not talking about lying down (though, that also has no exact opposite either, with “standing up” being just kind of an opposite). We mean the word for telling a lie. We don’t have a word for the opposite, a single word that means “tell the truth.”

As much consideration as I've given to words and language, I'd never thought of that, instead having to rely on a juvenile dick joke site to point it out to me.

To the limited extent of my knowledge, French is the same way.

Other words, meanwhile, have opposites that you’ve never heard of.

Or maybe we have, as in the case of "darkle" (or at least its poetic relative, "darkling"), but it's usually fun to go through these countdown lists anyway.

5 Forget Sparkles. It’s Time to Darkle

This appeals to me on a visceral level.

Darkling then gave birth to the word darkle. If something becomes concealed in the dark, it darkles.

And here I figured it was the other way around.

4 If You’re Not Ambidextrous, Be Thankful You’re At Least Not Ambisinistrous

Oh, but I am. Ever hear the expression "two left feet?" That's Waltz.

Having two of the same foot would ruin your dance skills, whether it’s two left feet or two right feet, but somehow, we never say someone has “two right feet.”

Hopefully, we all were already aware that "sinister" derives from a word meaning "left," revealing how important conformity is to society.

3 Euphemisms Smoothen the Conversation. Dysphemisms Make it Worse

If "dysphemisms" weren't already a word, we'd have made it up (well, all words were made up; it's just a matter of how long ago they were). Consider how they coined "dystopia" as the opposite of "utopia," for instance, the latter being a pun of "eutopia."

Over time, it’s quite possible for euphemisms to become dysphemisms. Originally, the words “moron” and “imbecile” were objective ways of categorizing someone by I.Q. When those mutated into insults, we came up with a euphemism that was more acceptable: retard. In time, retard became a dysphemism more offensive than any earlier synonym, and almost more offensive than any other word period.

I've noted this before, which is exactly why I don't accept the taboo around the word "retard." (Besides, it's a perfectly good word in French, albeit pronounced slightly differently, meaning simply "delay.")

We replaced "retarded" with "special," and now you can't call someone special without them thinking you're slyly calling them an idiot (which was another precursor to "retarded"). No matter what words we come up with for people in the slow lane, it will inevitably become a general insult, and then we have to think of something new and value-neutral, which will quickly become an insult in turn, and we have to find yet another word.

Sure, it would be nice (a word that used to mean "foolish") if we stopped insulting people altogether, but that ain't gonna happen.

2 Serendipity’s Great. Watch Out for Zemblanity

Okay, one I've never heard of or would have guessed.

George Carlin had a joke about how we call it a miracle when someone barely escapes death, but it’s actually just as miraculous when someone who’s almost certain to escape death dies anyway. We need a word for when the world seems to conspire so events pan out in just the worst way. Let’s call that zemblanity.

I've just been calling it "the normal state of things." Or maybe "exactly what I expect." Pessimism pays dividends on those rare occasions when things actually go your way.

1 Natural Gas Replaced Unnatural Gas

Except that nothing's really unnatural, but okay.

“Gas” refers to any number of substances that happen to occupy a certain state of matter. You’re currently surrounded by oxygen gas and nitrogen gas, for example (at least we hope you are, because if you’re not, you have just moments left). “Natural gas,” however, refers specifically to cooking gas, which consists mostly of methane, along with a few other trace ingredients.

I've known about chemistry from a very young age. I'm pretty sure I memorized the alkane series (methane, ethane, propane, butane, etc.) before I learned the alphabet. So it has bugged me almost my entire life that we (in the US) also use "gas" as an abbreviation for "gasoline," which is obviously a liquid (albeit a volatile one). I'd call it "petrol," but then people would think I'm trying to be British.

The reason natural gas got that name is it had a predecessor, which wasn’t nearly as natural. Back in the 19th century, we used to make gas — out of coal.

Which is also natural, but whatever.

I won't go into the cooked-up controversies surrounding natural gas use right now. Another time. This one's about words, and I've reached the end of mine for today.

4 Like • Comment

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