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

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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 ...
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The series of tubes that we call the Internet

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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 mea...
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