What are p-adic numbers? They are a different set of numbers first introduced by Kurt Hensel in 1897. The motivation at that time was an attempt to bring the ideas and methods of power series methods into number theory. They have been used in proving Fermat’s Last Theorem and have other applications in number theory. See http://mathworld.wolfram.com/p-adicNumber.html for more information.
A little terminology needs to be introduced. The p in p-adic represents any prime number. For each prime, there is a new and different set of p-adic numbers. Q2 identifies the 2-adics, Q5 represents the 5-adics, Q17 represents the 17-adics. To keep the same notation, Q∞ will represent the real numbers.
Another term to consider is “close.” The basic idea is that given a number n, it is close to 0 if it is highly divisible by a prime p. Consider the numbers 25 and 625. Relatively speaking, 625 has more factors of 5 than does 25, or in other words, 625 has a higher divisibility by 5. Therefore 625 is closer to 0 than 25. This idea will be made a little bit clearer in future postings.