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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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Photons, what are they?

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Probably everybody has once or twice in their lifetime heard the word "photon" together with the phrase "oh, a photon is just a particle of light," or something along those lines. That's all well and good, but what does "particle of light" actually mean? Can you touch it? How big is it? What does it look like? What can you do with it? And who cares? If someone talks about particles, you would expect them to mean that this thing is something solid, like a ping pong ball. You would also expect it to have some other properties that particles usually have, like it says on the wikipedia page about particles : " In the physical sciences, a particle (or corpuscule in older texts) is a small localized object to which can be ascribed several physical or chemical properties such as volume or mass. " Let's also take a look at what wikipedia says about photons : " A photon is a type of elementary particle, the quantum of the electro...
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