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

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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2 Like

Annette

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

Can I leave this comment?

July 26, 2024 at 11:39am

#1074411

Unbelievable?

I'll give Cracked a pass for using the phrase "scientific facts" like this, because they're a juvenile humor site, not a scholarly publication.

4 Scientific Facts Rational People Should Refuse to Believe
If we told you someone’s hiding edible microchips in your cheese, you’d think we were nuts

Mmmmmm... cheese and nuts...

Ahem. Sorry. Where were we? Oh, yeah.

We love telling you unbelievable facts. When we call a fact “unbelievable,” though, we still do expect you to believe it. You’re supposed to say, “I didn’t know this was true before, and it’s surprising, but I believe it now because I trust you.”

I trust you to make jokes. Sometimes those jokes are even funny. The idea that I'd trust this source for anything else is funny.

Note: We’re not telling you the following facts are untrue. Some of them definitely are true, and they’ll remain true whether you believe them or not.

As usual, skepticism serves well here.

4 Baby Animals Evolved to Look Cute So We’ll Protect Them

These cute baby animals share a specific set of characteristics, which we’ve dubbed kindchenschema. They have large eyes and small mouths.

Animators know this, which is why your favorite cartoon character protagonists have big eyes and small mouths. That much, I don't dispute. What I have a problem with is the reversal of cause and effect.

Also, the scientist who first described kindchenschema was a full-on Nazi.

Yeah, well, so was Heisenberg. That doesn't mean he was right or wrong about the science.

What if so many animals are born with big eyes and small mouths just because they’re still growing into their parts, rather than as a specifically evolved survival trait?

Okay, I can accept that hypothesis, but it doesn't address the cuteness factor: why do we find those traits adorable?

Look, Occam's Razor demands the simplest hypothesis, which is not that a diverse number of animal species began producing "cute" offspring to please their bipedal, mostly hairless overlords, but that said overlords themselves evolved to find those attributes "cute" because if people didn't find their own babies cute, there would be less reason to hang on to the whiny, poopy bastards.

I'm not saying that's correct, either. But it makes more sense.

3 Muscles Grow More When You Concentrate on the Mind-Muscle Connection

The fitness industry is full of new and exciting science explaining the right way for you to build muscle.

And most of it is bought and paid for by companies wanting to sell their latest torture devices, much like nutrition science is often bought and paid for by companies wanting you to consume their products.

One of the strangest factors here is something called mind-muscle connection. You will see more muscle growth if you think very hard about the muscle you’re exercising.

Okay, but that's not totally out there. The idea that mind and body are separate entities is obsolete. I'd just want to see real evidence, which I imagine is hard to come by.

What if this is a conspiracy, whose sole goal is to occupy your full attention when you exercise, so you’re oblivious to all else around you? This way, the conspirators can take sips out of your water bottle when you’re distracted, saving them from having to walk all the way to their own bottles, which are on the other side of the room.

See, now, that's funny.

2 They’re Putting Microchips in Parmesan Cheese

If you buy some parmesan cheese in the United States, the word “parmesan” on the label means basically nothing.

Well... it means it resembles cheese, but you can be pretty sure it doesn't resemble steak or an energy drink.

Europe is a different story. In Europe, that parmesan cheese you buy had better be the real thing, made in northern Italy.

One of these days, I'll figure out why this works for, say, champagne vs. sparkling wine, or tequila vs. mezcal / agave spirits, but not parmesan. But today is not that day. Tomorrow doesn't look good, either.

The blessed cheesemakers from Parma and Emilia-Romagna have a new tool to let you distinguish their wares from fakes: microchips.

Well, now, that's a reference joke that's growing increasingly obscure. It's from Monty Python's Life of Brian. Buncha peasants are standing around trying to hear the Sermon on the Mount, but they're pretty far away and microphones haven't been invented yet. So there are lines like "What'd he say?" "Blessed are the cheesemakers."

We’re going to find it hard to invent a conspiracy to explain this story, since our whole issue with it is it already sounds like a conspiracy. If we try, we’ll probably end up concluding that every other piece of food we eat also contains microchips, and the Parmigiano Reggiano ones only got so much attention because they got caught.

How about: we only think it's a conspiracy because the microchips we've absorbed from other sources control our minds to think so as a way to deflect blame?

1 Balls Hang Outside the Body to Keep Cool

Oh, hey, after the cheese entry, we're back to nuts. Yeah, not going to quote this one. I'll just point out that exposed balls are certain proof that men weren't intelligently designed. Jury's still out on women.

Which, of course, is the exact opposite of the author's joke at the end.

3 Like • Comment • View 4 comments

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