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DeutschExploding DotsP-adic Numbers
In the previous section, we managed to construct two non-zero N-adic numbers, for some integer n, are a special number system, with a definition of “closeness” that is different from usual arithmetic. Two n-adic numbers are close if their difference is divisible by a high power of n.
It turns out, however, that this problem only occurs if the number base is not a A prime number is a positive integer that has no factors other than 1 and itself. The first prime numbers are 2, 3, 5, 7, 11, 13, …
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 The prime factorisation of a number is a way to write it as a product of prime numbers. For example, the prime factorisation of 12 is Fermat’s last theorem states that the equation In 1637, the mathematician Pierre de Fermat made a note in a textbook, claiming that he had a proof, but that it was too large to fit in the margin. For over 350 years, other mathematicians tried to find this proof, and Fermat’s last theorem was even in the Guinness Book of World Records as the “most difficult mathematical problem”. In 1994, after many years of work, Andrew Wiles finally managed to prove it. He later received the Abel Prize, one of the highest honours in mathematics.
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 Paul Monsky (born 1936) is an American mathematician and professor at Brandeis University. He proved that a square cannot be divided into an odd number of equal-area triangles – this is known as “Monsky’s theorem”.
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.