# Category Archives: Uncategorized

## The isoperimetric inequality

Our story begins at around 800 BCE in ancient Greece with Dido of Tyre. As many stories from that time, this too started with a murder, when Dido’s brother Pygmalion killed her husband. Fleeing her brother, Dido eventually reached north … Continue reading

## The 15-Puzzle and the symmetric group

Almost 150 years ago the game 15-puzzle was invented and quickly gain lots of popularity. The rules are quite simple – you start with a board with 15 numbered pieces as in the image below, where the pieces are ordered … Continue reading

## Breaking up Pythagorean Triples

One of the thing I really love in mathematics is how different subjects, which at least when starting to learn mathematics seem far apart, can come together in interesting ways. While the first reaction of many people might be that … Continue reading

## Mediants in Mathematics

In the previous post we learned about the mediant, which is the “wrong” way to add rationals : . While this operation is not well defined, and depends on the presentation of the rational numbers and not just their values, … Continue reading

## On one weird rational average

One of the things that I love about Mathematics, is how one can come across a mathematical problem or a phenomenon which you can look at from several points of views, each one different, yet they all combine together to … Continue reading

## Counting primes in arithmetic progressions

We have now seen in the previous two posts a few results about prime numbers. In the first one, we saw why primes are so important and used Euclid’s proof to show that there are infinitely many primes. Moreover, we … Continue reading

## Summing the prime reciprocals

In the previous post we saw why primes are so important, and used Euclid’s proof to show that there are infinitely many primes. We further conjectured that not only there are infinitely many primes, they are also “nicely” distributed among … Continue reading

## Improper integrals and periodic functions

The idea for this post came from a question I saw in a math help forum about improper integrals. While this problem has a very simple solution using basic tools in integral calculus, I want to show a more geometric … Continue reading