I was reading about theorem called Monsky's theorem stating that "A square can never be divided into an odd number of traingles of equal area " . I read that the proof uses something called 2-adic valuation . I don't know anything about p-adic systems but know that "It is a slightly different number system where the proximity of two numbers is obtained by the number of times their gap is divisible by a certain prime number p " and unknowingly I consider p-adic to be advanced mathematical stuff . So , I was excited to find a possible instance where advanced mathematics being used in complicated problems of school geometry . Am I right in concluding that these 2-adic valuation is not same as real number systems as we know it and this is indeed one instance where higher maths is being applied to euclidean geometry ?
P.S: On a different note , how difficult a question would it be in a maths contest if it asks to prove " A square cannot be divided into odd number of triangles of equal area " ?
The geometry of $p$-adic numbers is very different from the geometry of real numbers. For instance, any triangle in $\mathbb{Q}_p$( set of $p$-adic numbers) is isosceles, which is clearly not the case for $\mathbb{R}$.