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 20 hours 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

a day ago
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?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

August 16, 2024 at 11:15am

#1075348

Cave People

With a Blue Moon coming up on Monday, this BBC article seems appropriate to talk about.

Cave discovered on Moon could be home for humans

It's a good thing it's the BBC, and not, say, the Daily Fail. Because the latter's headline would be something like "Cave discovered on Moon could be home to a space alien civilization"

I mean, that's technically true, for infinitesimally tiny values of "could."

Scientists have for the first time discovered a cave on the Moon.

We expected there would be caves, but it's good to confirm. Otherwise we show up on the moon with spelunking gear, only to find there wasn't a cave after all.

At least 100m deep, it could be an ideal place for humans to build a permanent base, they say.

This author's definition of "ideal" obviously differs from mine. To me, the ideal place for humans to build a permanent base is Hawai'i.

But, okay, I get that it's a suitable location on the moon.

In case you're wondering why caves are good:

Countries are racing to establish a permanent human presence on the Moon, but they will need to protect astronauts from radiation, extreme temperatures, and space weather.

They'll be useful on Mars, too, for the same reasons, plus dust storms. And don't forget meteoroid impacts. Those craters didn't form themselves.

Quibble: if you're going to live at a permanent Moon base (even temporarily), should you still be called an astronaut? They don't call the people living in Antarctica polar explorers.

Helen Sharman, the first British astronaut to travel to space, told BBC News that the newly-discovered cave looked like a good place for a base, and suggested humans could potentially be living in lunar pits in 20-30 years.

And reliable fusion power is 20-30 years away. Has been for 80 years.

But, she said, this cave is so deep that astronauts might need to abseil in and use “jet packs or a lift” to get out.

With gravity 1/6 that of Earth, I'm sure the clever loonies will think of something.

Lorenzo Bruzzone and Leonardo Carrer at the University of Trento in Italy found the cave by using radar to penetrate the opening of a pit on a rocky plain called the Mare Tranquillitatis.

It is visible to the naked eye from Earth, and is also where Apollo 11 landed in 1969.

Today, in Pronouns with Ambiguous Antecedents: Mare Tranquillitatis is visible from Earth (it's one of the big dark splotches). Not the pit.

It was made millions or billions of years ago when lava flowed on the Moon, creating a tunnel through the rock.

I'm pretty sure scientists know whether it was "millions" or "billions." Also, I wasn't aware that the Brits started using "billion" to mean a thousand million, like Americans. It used to mean a million million there. What we call a billion was called a milliard or something, because all those terms were borrowedstolen from French, likely because the Brits didn't have enough fingers to count that high.

Once Prof Bruzzone and Prof Carrer understood how big the cave was, they realised it could be a good spot for a lunar base.

“After all, life on Earth began in caves, so it makes sense that humans could live inside them on the Moon,” says Prof Carrer.

I realize it's in a quote and not the author's words, but that sentence has so much wrong with it that I hardly know where to begin.

That could open the door to finding evidence of life on Mars, because if it did exist, it would almost certainly have been inside caves protected from the elements on the planet’s surface.

I'm not sure about that one, either. Is the author talking about finding fossilized evidence of life on Mars? Because that could very well be on the surface, especially in places where water once covered. Existing life on Mars is still entirely speculative, but I'd agree that if it did exist, it'd be underground.

In any case, I find it amusing that we think of early humans as "cave people," and here we are making plans to encave ourselves on other worlds.

2 Like • Comment

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