# Advanced

Posts that assume some familiarity with advanced math concepts (e.g. integrals, set theory) and/or contain proof that require some mathematical maturity to understand well

## Arrow’s Theorem (or: How I Learned to Stop Worrying and Love Dictatorships)

Elections never really feel fair (or, if talking about the electoral college, are never fair) Here is an example, say there is an internal election in some political party, to determine the order of members of the party. To do so, each voter votes for their favorite party member, and the party members are ranked … Read more

## The Paradox of Periodicity (of Functions)

You know periodic functions right? Functions that after a certain place start repeating, for example sin(x): More formally, a function f is called periodic is for some constant p>0, f(x+p)=f(x). In other words, shifting by p does not affect f(x). The constant p will be called the period of the function (sometimes we say f … Read more

## Why does this fraction give the Fibonacci sequence? It’s no coincidence

You may have seen one of the following viral math facts: The first fraction seemingly gives the Fibonacci numbers. That is, the sequence of numbers that starts with 1, then 1, then each successive term is the sum of the two prior terms. So the first few Fibonacci numbers are 1,1,2,3,5,8,13,21,… Which appear on the … Read more

## The Spooky Function Horror Show

Be prepared, for you are about to see some of the scariest functions this Halloween. The excerpt is like this to not spoil what’s within!

## A Devious Bet: The Borel-Cantelli Lemma

Suppose you are offered a bet as follows, by a totally not evil and totally not a squid monster disguised as a man casino owner. The bet will have (countably) infinitely many steps. In each you win or lose money, the only thing the totally not a squid monster tells you is that in each step, on average, you gain money. Furthermore, that average increases as the bet goes on. Do you take it?