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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 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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