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Exploding DotsP-adic Numbers

Reading time: ~10 min

In the previous section, we managed to construct two non-zero 10-adic numbers M and N, so that M×N=0. This means that it is impossible to divide by numbers like 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 prime number. Since 10 , the 10-adic numbers are flawed. 2-adic or 3-adic numbers, on the other hand, are not.

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 prime factorizations. Since 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 Fermat’s Last Theorem.

One of the must surprising applications of p-adic numbers is in geometry. Here you can see a square that is divided into ${2*x} small triangles of equal area:

As you move the slider, you can see that it is possible to divide the square into any number of equal triangles.

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 Paul Monsky – you can even have a look at the paper he published about it:

The American Mathematical Monthly

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.