Posts

Showing posts with the label Study materials

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 ...

Entropy and the arrow of time

Image
When you whisk an egg and leave it alone, why doesn't the egg white and yolk separate on their own? If you light a fire, why does it keep burning? And why do we perceive that time never changes it's direction? All of these questions are answered by a single concept in physics: entropy. Simply put, entropy is a measure of the microscopic disorder of a system, and the second law of thermodynamics says that it can only increase or stay constant in a closed system. And all of this is just simple statistics. Let's put this in more concrete terms with a simple example: we flip three identical coins and record the outcome. First of all, what does this have to do with anything? In thermodynamics, we are interested in the energy of the studied system. That energy is made up of all of the individual energy states of the particles that make up the said system. The two sides of a coin represent the possible states of a two level system, which makes it quite a popular example. ...