Dirichlet's approximation theorem

Imagine you have some favorite toys in a box, but you want to make sure that you have enough of each toy to play with. You don't want to leave any toys behind!

Dirichlet's Approximation Theorem is like trying to make sure you have enough toys to play with.

Instead of toys, it's about numbers. These numbers are like the toys in the box.

Dirichlet's Approximation Theorem says that no matter what number you pick, you can always find two other numbers that come really, really close to it.

It's like if you had a toy box with a bunch of cars and one toy train. If you really wanted to play with the toy train, you could pick two cars that look similar to the train and imagine they were the same thing.

In math, this is called an approximation. Instead of having the exact number you want, you can use numbers that are really close to it.

Dirichlet's Approximation Theorem is important because it shows that you can always do this, no matter what number you're working with!