Partially Coherent

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 ...

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...

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...

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 ...