Is 0.999... = 1? (spoiler alert: no it is not)

You may have encountered the popular claim that \( 0.999... = 1 \), where the three dots signify that the decimal continues forever. This is a somewhat weird claim, since it would mean that mathematics is broken. There should be no way for two different numbers to have the same value. What makes it weirder is that this is quite popular claim. I've even seen mathematicians say that it's true! But is it though? One popular proof is to first denote \( S = 0.999...\) and then multiply by \(10\) to get \( 10S = 9.999...\) and subtract \( S \) from it, to get  \( 10S - S = 9.000...\) and finally dividing by \(9\) yields  \( S = 1.000... = 1 \) and we see that  \(0.999... = 1\)! However, there's a problem. This short derivation is not strictly speaking correct. It is veeeery close to being correct, and to see why let's look at finite decimals first. Let's say that \(S = 0.999\) (note that this is not the same as \(S = 0.999...\) ). Let's do the same trick as before, so

The series of tubes that we call the Internet

Maybe some people still remember the infamous statement made by the late Republican Senator Ted Stevens during the net neutrality debates in 2006, that the Internet is "a series of tubes." He used this statement to criticize the bill on net neutrality, because according to him, the Internet is not a "big truck" and tubes can get clogged. After the statement he was met with ridicule, because of a very limited understanding of the Internet, even though he was in charge of its regulation.



To me, the extreme criticism that Stevens got for the tube statement is a little weird, because let's face it, the Internet is, in all actuality, a series of tubes. And no, I don't mean in a metaphorical sense. Although, with the word tube, I don't mean plumbing pipes or whatever.
I have no idea what Stevens meant with tubes, and I don't think that he made a good argument against net neutrality. The argument was just ridiculed for all the wrong reasons. I mean come on, he said that "an Internet was sent by my staff at 10 o'clock in the morning on Friday. I got it yesterday [Tuesday]." Well, if he was sent an Internet, I would expect that to take some time! And as for the part where he says "the Internet is not something that you just dump something on. It's not a big truck," he is comparing information that is literally moving at the speed of light, to trucks.

But anyway, back to the point. The Internet we have today is formed by a vast network of optical fibers. You can think of an optical fiber as a kind of "tube," because it functions exactly like one. Both fibers and tubes are just paths, along which things can flow. In optical fibers, the things that flow are pulses of light, and in plumbing pipes... Well, its something else. And as I mentioned in an earlier post, the modern internet would not be possible without optics. But why is this?

You might think that if we didn't have optical fibers, we could just connect everything with electrical cables, and there, problem solved. Since we have a cable for every job, such as the USB, HDMI, DP, ethernet, coaxial, and so on, then surely we have an electrical cable that could replace the optical fiber, right? The cable that has the highest potential to do this is the coaxial, but it has two large limitations: 1. the coaxial cable is not capable of ultra high bandwidth transmission on par with the optical fiber and 2. it is made of copper, which makes it expensive. In other words, if the Internet was based on coaxial cables, it would be really slow and reeeally expensive. The two things that people don't want to hear in the same sentence with Internet, oh dang it.

What makes optical fibers so special, is that they are cheap to produce, have very low losses, and information can be sent in the form of optical pulses, which travel at (unsurprisingly) the speed of light. Those pulses are being held in check by a phenomenon known as total internal reflection. This effect causes light to get confined inside a material that is optically denser than its surroundings, given that the angle of incidence is larger than the critical angle. For example, glass and water are more optically dense (high refractive index) than air (low index). Total internal reflection is quite easily achieved, and one of the coolest demonstrations of this effect is light that is guided by a stream of water, as shown here by James Dann.



In order to connect the world, these fibers have been laid all over, even across the Atlantic as undersea cables. The fibers themselves are really small, the core that guides the pulses can be just 9 micrometers in diameter (a human hair is about 30 to 100 micrometers) and there are usually about six of those in one cable. But all of the protective layers on top make the whole thing quite hefty, about the size that fits snugly in a sharks mouth. Oh right, sharks like to chew on them for some reason, which is a major problem.


Sharks are not the only issue, believe it or not. In my Master's studies we had a course on optical telecommunications, and back then, the professor would start the course by saying "the optical fiber features nearly unlimited bandwidth!" Now, just a few years later, I talked with the same professor about this topic again and he told me that he no longer says that in the course. We have almost reached the bandwidth limit of the optical fiber, even though we have used almost every trick in the book. We need to find some new bits if we want to increase the bandwidth further!



As a side note on net neutrality (US citizens, pay attention!), the whole point of that particular law is to keep corporations from making their own rules; a classical user vs. provider situation. That is, net neutrality is one government regulation that staves off nearly an unlimited amount of corporate regulations, and now the US seems to have decided that they prefer the latter.

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