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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QPdoll

about 15 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 18 hours ago
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Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
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WakeUpAndLive️~Happiness

a day ago
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Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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August 3, 2023 at 10:45am

#1053583

Warped Minds

Y'all should know by now that I'm a Star Trek fan. Not enough to dress funny and go to conventions, but I appreciate the various shows and movies (mostly).

But despite comments like the one I made the other day about Zefram Cochrane inventing the warp drive, I know it's fiction, mythology, and metaphor, not future reality. For one thing, I doubt very much that there are other tech-capable species in our vicinity (and there are almost certainly none that look like us but with forehead bumps), and for another...

Is Star Trek’s Warp Drive Possible?
The concept of the warp drive is currently at odds with everything we know to be true about physics.

And yes, I know that our understanding of physics is evolving and that lots of things that used to be considered impossible, aren't. So while some things are definitely impossible (counting to infinity, e.g.), I hesitate to give technological advancements that label.

Central to science fiction, and Star Trek in particular, is the ability to travel the galaxy at speeds far faster than light via a fictional technology called a “warp drive.” What is that, and will we ever have one?

Trek didn't come up with the concept; it's been around in various forms in SF for probably half as long as there's been SF (which roughly corresponds to the time frame in which we've known that matter cannot be accelerated to or past the speed of light).

A provisional answer is “no.” According to the accepted laws of science, nothing can travel faster than light.

That is, based on my admittedly limited knowledge, an oversimplification at best. But close enough for our purposes.

The distances are so vast that interstellar travel would take lifetimes.

Not if we extend our lifetimes by uploading our consciousnesses into machines... another thing that may not turn out to be possible.

That makes for a boring sci-fi plot, so the creators of Star Trek invented a convenient, but imaginary, technology.

Again, they didn't invent it, even conceptually. Early shows weren't even consistent in how it was supposed to work. They did, arguably, popularize it, though.

According to Star Trek canon, the warp drive works by creating a “warp bubble” around the spaceship, inside of which space is literally warped. In front of the spaceship, space is compressed, while behind the vessel, it is expanded. In this way, a spaceship never travels faster than light; it merely passes through a shorter distance.

The difference between "traveling faster than light" and "passing through a shorter distance" really needs more explanation here, but the article couldn't be arsed, and neither can I. Plenty of information out there, though, if you look for it.

Is this realistic? Maybe.

"Maybe" does a lot of heavy lifting here.

In 1994, theoretical physicist Miguel Alcubierre found a solution that distorted space in a way very similar to that originally envisioned by the creators of Star Trek. Under exactly the right conditions, it is possible to expand space behind an object and compress it in front.

The Alcubierre Drive is something I've seen stuff about before. There's still a difference between something being theoretically possible, and actually attaining it. For example, from what we know about quantum mechanics, it's theoretically possible for an object of arbitrary size to suddenly appear seemingly out of nowhere. The chance of this happening, though, would be akin to that of an individual winning a fair lottery... every day for a billion years.

If the universe really is infinite, though, this has happened. Hell, it might be the actual origin of the universe. Doesn't mean we can make it happen.

There is a problem, though. To accomplish this distortion, researchers would have to use negative energy—that is, reduce the energy of empty space to below zero. According to Einstein, a warp drive requires an impossible premise.

That, the negative energy bit, is definitely a serious problem. That doesn't excuse the article's appeal to authority. Einstein was right about a great many things, but we don't just take his word for it; we do experiments to back up the theory. He was human; he was wrong about a great many things. Just ask his wife.

Therefore, while scientists try to find loopholes in the conditions required for Alcubierre’s solution, most think that a warp drive will not be created this way; negative energy is a mathematical artifact and not a physical phenomenon.

On the flip side of that, for a long time, people considered negative numbers to be mathematical artifacts and not representative of physical phenomena. It took a paradigm shift to start thinking of them as "real" numbers. And don't even get me started on that extension of negative numbers, complex numbers.

Altogether, our current understanding of the laws of nature neither allows for faster-than-light travel nor an Alcubierre-like solution to warping space. However, those who dream of traveling the stars should not give up. Scientists do not have a full understanding of the laws of nature.

So yeah, I'd hesitate to call it impossible. Within the timeframe of the Star Trek series? Unlikely. Though probably not as unlikely as that other Trek mainstay, the transporter.

But that's a story for another time. I have a new episode of Strange New Worlds to watch.

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