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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 before, so
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How large is infinity?

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If you are like most people, then you have probably heard children arguing over some kid thing. Have you ever been astonished how easily they resort to infinity? It's like these kids don't even know how large it really is! But how large is infinity? Let's start looking into it with some actual numbers that you can write down in some form. The first really big and easy to understand number is googol . You can google it, googol is a real thing, and Google is actually a misspelling of googol. (Fun fact: the study of very large numbers is called googology.) Googol is simply \(10^{100}\). That's 1 followed by a hundred zeros, so yeah, it's quite a large number. One way to illustrate the size of googol is to compare the mass of the electron, which is \(10^{-31}\) kg, to the estimated mass of the universe, about \(10^{50}\) or \(10^{60}\) kg. If you take one googol of electrons, their mass will be up to \(10^{19}\) times larger than the mass of the whole un
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