Exploding DotsP-adic Numbers
In the previous section, we managed to construct two non-zero
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
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.