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Showing posts from February, 2018

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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No, physicists did not create a new form of light

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Last Friday a collaboration between MIT and Harvard made the headlines with their research on quantum photonics, and I must say, the hype is particularly strong with this one. Here's my pick of the headlines from some of the popular science outlets: "Physicists Created a New Form of Light" - Motherboard "Photons entangled to make new form of light" - New Atlas "MIT Physicists Have Constructed a Bizarre Form of 'Molecular' Light With 3 Photons - photon's shouldn't do this" - ScienceAlert "Scientists create a new form of light in breakthrough that could pave the way for everything from ultra-fast quantum computers to real-life LIGHT SABERS" - Daily Mail I cannot overstate how badly overhyped this study is. First of all, the team did not find a new form of light, and no entanglement was involved. Neither did they make "molecular" light and photons are supposed to do exactly this. Oh, and you definit...
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Sum over all natural numbers is...

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There are some really weird things in mathematics that seemingly don't make any sense. And sometimes, even if you are explained why things are the way they are, you are still left with a boggled mind. One of those things is the deceptively simple question of what is the sum over all natural numbers, that is, the sum 1+2+3+... all the way up to infinity. One might say that this question is completely trivial, but what if I were to say that the sum is exactly -1/12? The first reaction is usually that it doesn't make any sense, but a simple proof of this is explained in one of Numberphiles most popular videos, titled "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12" which you can see below I can assure you that the Numberphile proof is completely fine, although they don't go to details. But this only raises new questions rather than answers existing ones. We are summing positive whole numbers together, so how can the result be a negative fraction? If you ...
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