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Showing posts from January, 2019

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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Forced correlations & In search of lost rationality, part 2.

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As promised, I'm back for some more lost rationality! And ranting. Yeah, mostly ranting. But before I go there, I'll lay out the research that this rant is related to. This one was about temporal coherence, where we look at correlations between different points in time instead of space. To measure those correlations, you need a device such as a  Michelson interferometer  where you split the incoming light to two, delay one of the copies, and then recombine them. This simple setup is sufficient only for some special cases, and measuring the correct correlations is way more complicated. There actually isn't a general method to do that, modern detectors are far too slow! Let's just say that if you need to measure the temporal coherence of a pulse train, then one way to do it is to isolate individual pulses and measure them with a  FROG  (yes, that is a real scientific instrument) and then study their correlations from the measured data. Anyhow, if you have a de
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