The p-adic absolute value gives us a new way to measure the distance between two numbers. The p-adic distance between two numbers x and y is the p-adic absolute value of the number x-y. So going back to the 3-adics, that means numbers are closer to each other if they differ by a large power of 3.

In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

A -adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime are related to proximity in the so called ” -adic metric.”

How do you calculate p-adic expansion?

The proof of Theorem 3.1 gives an algorithm to compute the p-adic expansion of any rational number in Zp: (1) Assume r < 0. (If r > 0, apply the rest of the algorithm to −r and then negate with (2.2) to get the expansion for r.) (2) If r ∈ Z<0 then write r = −R and pick j ≥ 1 such that R < pj.

Here’s a final curious fact about the p-adic numbers. We all know that if x and y are two non-equal real numbers then either xthere is no linear ordering of the p-adic numbers!

mathematician Kurt Hensel
The p-adic numbers were invented at the beginning of the twentieth century by the German mathematician Kurt Hensel (1861–1941). The aim was to make the methods of power series expansions, which play such a dominant role in the theory of functions, available to the theory of numbers as well.

What is P in number theory?

In set theory, P(X) means the power set of X. In geometry, P can refer to a projective space although it is also often the name given to a particular point. 9K views. · Related questions (More answers below)

What is the meaning of ADIC?

Acronym Definition