We all know about Pythagoras and his obsession with triangles. Usually when we first learn about the rational and real numbers, we are also told about the Pythagoreans, a 6th century BCE cult that started with the followers of Pythagoras and which dealt with mysticism and Mathematics (and if it still exists and you are a member in it, please send me a message – I would love to join). The legend says that they loved ratios between natural numbers, i.e. rational numbers, and didn’t believe that there exist any other type of numbers. Unfortunately, Hippasus which was one of their own “betrayed” them and proved that is not rational. When he died by drowning, the Pythagoreans said that this was a punishment by the gods themselves for showing that the world of Mathematics is not perfectly rational. Well, it was either this, or because he built a dodecahedron inside a sphere. Anyway, these days we know that there are irrational numbers, of course, but it is nice to know that if you insult the gods of Mathematics you will be punished.
The moral from this story is that rational numbers are divine, but there are other types of numbers – like the real numbers. While there are irrational numbers, there are so many rational numbers so that we can get as close as we want to any real number. In this post we will try to investigate when should we say that an approximation is good, how good of an approximation can we hope to find for a given real number, and what is the connection to lattices (see the post about lattices for the definitions and some examples).