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:

Write_Mikey_Write!

about 3 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 4 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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').

1 Like

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.

April 7, 2021 at 12:04am

#1007916

Ain't Goin' Nowhere

And now to crush all your dreams...

Why interstellar travel is so damn hard
Basic physics goes a long way towards explaining why a pair of theoretical rocket engines may never come to fruition.

Rocket science is easy. Newton figured that shit out. Action. Reaction. F=ma. There, done; what made "rocket science" a synonym for supergenius?

Well, it turns out there's a lot of details to work out. Who knew?

Ever since we imagined the technological sophistication to send ourselves hurtling at escape velocity away from the Earth and toward some unknown pinprick of light in the unending vacuum of open space, obsessive individuals and paranoid governments have spent billions of dollars trying to figure out how to get as far away from our home planet as possible.

That's a cynical way to look at it. The idealist version is we're curious and want to explore. Need to explore, actually.

As for the next big thing, the general idea that the American government has run with for the past 70 years—from Project Orion to Project Longshot to Project Prometheus—has involved strapping small nuclear bombs to the backs of rockets and hoping for the best.

It occurs to me that some people might not have heard of these things. I forget that not everyone reads science fiction. Yes, there have been serious proposals to use nuclear bombs for rocket thrust. Look it up if you don't believe me.

Needless to say, none of these have ever been built or are likely ever to be built; they were dreamed up with a future civilization in mind, one that had conquered the problems of nuclear fusion and international cooperation but still had no control over the Earth’s inevitable solar doom

That's rich. My turn to be cynical: Should we survive all the looming catastrophes such that we have to start worrying about the Sun heating up and destroying Earth (not something to worry about for, oh, half a billion years at least, so don't start sweating yet), we will figure out a way to engineer the Sun before we figure out international cooperation.

As long as there are some people who think pineapple is just fine on pizza and others who think it's an abomination, there will always be war and conflict.

If you’re wondering why taking a trip to another star is incredibly difficult, blame physics. Conservation of momentum (or Newton’s third law, depending on how you want to look at it) requires a rocket to poop out some amount of mass at some speed (AKA explosive fuel) for the rocket to move. The sticking point is that the fuel still has to push the remaining fuel still unpooped and connected to the payload.

Gotta love those technical terms.

This predicament can be turned into a formula that relates the change in speed to the amount of mass pooped out. It’s called the Tsiolkovsky rocket equation, named after the father of modern rocket science.

This is the same Tsiolkovsky who, after looking at all the math needed to lob rockets into space (he lived about 100 years ago), said "fuck it, what if we built a really tall tower?" Only he said it in Russian so no one listened.

It can tell you that if you have a chemical propellant and you’re going to eject your fuel at, say, the maximum velocity of a nuclear fireball — around 100 km/s — and you want to travel 4.25 light-years over to Proxima Centauri, you’re going to need to have ten thousand trillion trillion trillion trillion times more fuel than payload if you want to get there in around one hundred years. Not to mention that we’d need double the fuel and time to slow down enough to take data from or drop passengers near the star. For a 1 kg payload, the fuel would roughly account for the entire mass of the universe.

This is the kind of shit that space-based science fiction loves to hand-wave around, to the point where people are all like, "Why don't we just invent warp drive?" Like that's easy. I mean, sure, maybe? We're actually pretty clever when we try to be. But we barely have the vaguest glimmerings of a theory of how such a thing would be possible. Meanwhile, it's a plot device, nothing more.

The sheer weight of the scientific difficulties facing interstellar travel is humbling, if not existentially depressing. Many space nerds have put their hopes in harnessing exotic engines that utilize badly understood (or entirely misunderstood) physics that might obviate the fuel problem. Two of them, the EmDrive and the Mach Effect Thruster, have been hyped by everyone from NASA to National Geographic as the solutions to our interstellar detention. It’s too early to say if either of them is a pipe dream, but their tantalizing likelihoods fade every day.

The article goes on to describe these technologies, mostly in plain language (what do you expect from an article that calls rocket exhaust "poop?") so it's worth reading; I won't copy much more of it here.

For his part, Shawyer seems to be doing great: after receiving a patent for the EmDrive in late 2016, he formed a joint venture with some guys who have been trying to make flying cars happen for more than a decade.

AKA "doing holy work."

Before you comment with links, yes, I know there are flying car prototypes out there. I don't care. I want my consumer-model flying car. It's 2021 and I still don't have a flying car. I was promised one long before now.

For the foreseeable future, we will have to settle for a less glamorous, less hopeful kind of interstellar travel.

Still, there's always the possibility of new breakthroughs in physics. Like I said, we can be clever. History is littered with ideas that just waited for technology to catch up, and others that seemed impossible until someone made them possible. Unfortunately, it's also littered with bullshit, but we tend to forget things like "riding a dew-powered ship to the Moon" (which was actually the basis for one of the earliest "trip to the moon" stories) in a kind of collective confirmation bias.

So I'm not saying warp drive can never happen. And we certainly shouldn't stop imagining what we'll find when we finally have a way to head out there.

Mini-Contest Result!

Lots of great comments from yesterday! Some of them I might actually try. And I appreciated all the comments; some of y'all are overachievers. I said I'd pick one winner, though, and that's going to be...

Sharmelle'sThankfulExpressions

for her Bacon, Sausage, Runny Yolk, Hashbrown Sandwich

While I'm not actually a fan of runny yolks, it's trivial to simply cook the egg more thoroughly. And I do love me some breakfast. So a MB will be sent your way soon! But like I said, that wasn't the only one I liked, so thanks again for the ideas! I'll do this again sometime soon, so there'll be more chances.

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