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Showing posts from November, 2018

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

Quantum mechanics is broken

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Quantum mechanics, the brainchild of some of our greatest scientific minds, is broken. Some may object to this, since the theory is one of the most successful ones we have ever had. Indeed, it gives correct results, but that doesn't mean everything is okay. Let me illustrate with an analogy. Imagine you are driving along and suddenly the check engine light turns on (or whatever indicator your car has). But everything seems to be working fine, so you just keep on driving, although the on-board analytics is trying to show that something is wrong. And you just keep on driving, hoping that it doesn't blow up. You can't know how terribly wrong things are before you take the car apart and look inside. I am in no way saying that there is necessarily something wrong in the results of the theory. What I am saying is that although we have every indication that there is something wrong with it, we keep on using it. We've been ignoring the quantum check engine lig