# Exploding DotsP-adic Numbers

In the previous section, we managed to construct two non-zero *M* and *N*, so that *M* and *N* – a serious flaw in any number system.

It turns out, however, that this problem only occurs if the number base is not a

Mathematicians call these numbers ** p-adic numbers**, where the

*p*stands for “prime”. Even though they don’t seem particularly relevant in everyday life,

*p*-adic numbers turn out to be very useful in certain parts of mathematics.

For example, many unanswered problems in mathematics are related to prime numbers and *p*-adic numbers were defined using *multiplication* rather than *addition*, they are perfect for analysing these problems. *P*-adic numbers were even used in Andrew Wiles’ famous proof of

One of the must surprising applications of p-adic numbers is in geometry. Here you can see a square that is divided into

As you move the slider, you can see that it is possible to divide the square into any

But what about *odd* numbers? Draw a square on a sheet of paper, and then try dividing it into 3, 5 or 7 triangles of equal area.

Here’s the shocker: it turns out that it is *impossible* to divide a square into an *odd* number of triangles of equal area! This was proven in 1970 by mathematician

In the proof, Monsky had to use the 2-adic number system. Mathematics, no matter how abstruse it might seem, always comes up with surprising and unexpected applications.