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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March 29, 2022 at 12:01am

#1029660

Beam Me Up

I have a few articles in the queue related to science fiction technologies popularized in Star Trek. This is one of them.

Why Can’t People Teleport?
Set your phasers on stun, because we are going to beam you up on the physics of teleportation.

Sadly, none of them involve phasers. Yet. If I find one, it'll pop up here eventually.

Let’s face it: Nobody likes to travel.

Wow. You're not playing to me with that lede, are you? I love to travel. I'm the avatar of the phrase "It's not the destination, but the journey." I've even been known to say that there is no such thing as destination, only the journey.

I make an exception for airport security theater and steerage seats on airplanes. But everything else about travel, in any mode, is part of the adventure to me.

Still, I could see the advantages of the topic, which is teleportation.

Whether they’re traveling to get to an exotic location for vacation or traveling to work on a daily commute, nobody actually likes the part where they have to travel.

I'm the exception that negates the rule.

The people who say they like to travel probably mean they like to arrive. That’s because being somewhere can be really fun: seeing new things, meeting new people, getting to work sooner so you can go home early and read physics books. The actual traveling part is usually a drag: getting ready, rushing, waiting, rushing some more. Whoever said “it’s the journey, not the destination” clearly never had to sit in traffic every day and never got stuck in a middle seat on a transatlantic flight.

This article could have skipped these first few sentences and I'd be more inclined to feel sympathetic to the author.

Teleportation has been a fixture in science fiction for well over 100 years.

And I'm pretty sure some version of it was in fantasy long before then.

We could get to other planets more easily, too. Imagine sending colonists to the nearest habitable planet (Proxima Centauri b, four light-years away) without having to spend decades in transit.

Goddammit, how many times do I have to say this: there is not the slightest speck of evidence that any planet orbiting Proxima Centauri is "habitable" by our definition, and plenty of evidence that they're not. You're losing me here.

But is teleportation possible? And if it is, why is it taking scientists so long to make it a reality? Will it take hundreds of years to develop, or can I expect it as an app on my phone sometime soon?

I think I have a good idea of the answers to these questions, but I'm going to read on anyway, my annoyance with the writing notwithstanding.

If your dream of teleportation is to be here in one moment and then be in a totally different place the next moment, then we are sad to tell you right off the bat that this is impossible.

Unfortunately, physics has some pretty hard rules about anything happening instantaneously.

Okay, but look: any speed-of-light transmission from one point of the Earth's surface to another is close enough to instantaneous as to not make any practical difference. Even hypothetically bouncing a signal to geosynchronous orbit and back would have a lightspeed delay measured in seconds. We're splitting hairs here.

Information has to travel through space just like everything else, and the fastest anything can travel in this universe is the speed of light. Really, the speed of light should have been called the “speed of information” or “the universe’s speed limit.” It’s baked into relativity and the very idea of cause and effect, which are at the heart of physics.

All of this is true, but so what? We're supposed to be talking about teleportation, not that other Trek-popularized technology, warp speed (or FTL travel). That's a different discussion. The speed of light -- or information, whatever -- has little to do with whether we could build a transporter or not.

Still, I will grudgingly admit that this part of the article is good for people who need a primer on the universal constant known as the speed of light.

If that’s the case, then there are two options for making a teleportation machine work:

Your teleportation machine could transmit you to your destination at the speed of light.
Your teleportation machine could somehow shorten the distance between where you are and where you want to go.

Option No. 2 is what you might call the “portal” type of teleportation. In movies, it would be the kind of teleportation that opens up a doorway, usually through a wormhole or some kind of extradimensional subspace, that you step through to find yourself somewhere else.

That's more associated with the Stargate series, though Trek has certainly used wormholes as plot devices.

Sadly, both of these concepts are still very much theoretical.

By which is probably meant, "there have been no practical demonstrations."

Much more interesting to talk about is Option No. 1, which, as it turns out, might actually be something we can do in the near future.

And I think that's optimistic, analogous to how practical nuclear fusion has been 20 years away for the last 60 years.

To do that, you might imagine a machine that somehow takes your body and then pushes it at the speed of light to your destination. Unfortunately, there’s a big problem with this idea, and it’s that you’re too heavy. The truth is that you’re too massive to ever travel at the speed of light.

This is true for some of us more than others.

But seriously, though, that's an oversimplification. Photons can only move at the speed of light, for example. They have no mass. Particles with mass cannot move at the speed of light, only below it.

I mean, the article does say this, but I feel like some of it may be misleading.

But does that mean teleportation is impossible? Not quite! There is one way it can still happen, and that’s if we relax what “you” means. What if we didn’t transport you, your molecules, or your particles? What if we just transmitted the idea of you?

And this is where things start verging on metaphysics.

Here’s a basic recipe for speed‑of‑light teleportation:

Step 1: Scan your body and record where all your molecules and particles are.
Step 2: Transmit this information to your destination via a beam of photons.
Step 3: Receive this information and rebuild your body using new particles.

I mean, that's kind of how Trek handwaves the technology. Though I think they pretend the particles themselves are somehow converted to photons and then back again at the destination.

Oddly, that is not the most questionable science in Star Trek.

So it’s not hard to imagine that one day we might be able to scan and then print whole bodies.

No, it's not hard to imagine -- until you really think about the amount of information being processed.

The real limitation, though, might not be technological but philosophical. After all, if someone made a copy of you, would it actually be you?

I'm not going to say this isn't a legitimate discussion to have. And people have talked about it. The problem is, as with cloning, first you have to agree on precisely how the technology works. And since we don't yet have the technology, the discussion remains in the realm of ideas.

At base, though, I think any such discussion ends up getting mired down in the question of what exactly consciousness is. It's not the physical particle replacement that's at issue, in my opinion -- none of us contain the same particles we did when we were born (except maybe tooth enamel, if you're lucky enough to still have some) -- but the matter of consciousness, which we might not solve until after we figure out teleportation, warp drive, and synthehol.

Anyway, the article does delve into these questions somewhat, and at a level geared toward people without a lot of science background. I just wouldn't consider any of it as settled science.

After all, how much can you believe someone who starts out asserting that the destination is more important than the journey?

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