This is a promissory Maple package, which is rarely used (I found nothing in MaplePrimes and in Application Center.). Let us see the ?padic package. It is well known that the field of rational numbers Q is not complete. For example, there does not exist a rational number k/n such that k^2/n^2=2. There are only two ways to complete Q ( http://en.wikipedia.org/wiki/Ostrowski's_theorem ) . The first way is to create the field of real numbers R including Q. Every real number can be treated as a decimal fraction sum over [k in K] of a[k]*10^(k) with a[k] in {0,1,2,3,4,5,6,7,8,9}, finite or infinite. For example, the numbers 0.3+O(0.1), 0.33+O(0.01), 0.333+O(0.001), 0.3333+O(0.0001), ... approximate the number 1/3.

The second way is as follows (see http://en.wikipedia.org/wiki/P-adic_number for more details). We choose a prime number p and consider the valuation v[p] of a rational number k/m=p^n*a/b <>0 where integers are supposed to be irreducible :v[p](k/m):=p^(-n) , v[p](0):=0. The completion of Q up to this valuation is the field of p-adic numbers Q[p] (also including Q). Every p-adic number can be treated as a p-adic fraction sum over[k in K]of a[k]* p^(k) with a[k] in {0, 1, 2, 3, p-1}. For example, the numbers 2, 2+O(5),2+3*5+O(5^2),2+3*5+5^2+O(5^3) approximate the number 1/3 in Q[5]. These can be obtained with Maple as follows.

> with(padic);

> evalp(1/3, 5, 1);

2

> evalp(1/3, 5, 2);

2+O(5)

> evalp(1/3, 5, 3);

2+3*5+O(5^2)

> evalp(1/3, 5, 4);

2+3*5+5^2+O(5^3)

The field Q[p] is a very strange object. For example, the set of integers is bounded in Q[p] because v[p](k) <= 1 for every integer k. Another striking statement: the sequence p^n tends to 0 in Q[p] as n approaches infinity. The functions expp(x), logp(x), sqrtp(x) and the others are defined in the usual way as the sums of power series (see ?padic,functions for more details). For example,

> Digitsp := 12;

> logp(2+3*5+5^2, 5);

5+5^2+4*5^3+5^4+3*5^6+4*5^8+3*5^9+5^10+3*5^11+O(5^12)

> cosp(x, p, 2);

padic:-cosp(x, p, 2)

> eval(subs(x = 0, p = 5, padic:-cosp(x, p, 2)));

1

> eval(subs(x = 3*5, p = 5, padic:-cosp(x, p, 2)));

1

The definition of the limit of a sequence in Q[p] is identical to the one in R (of course, abs(x[n]-a)<epsilon should be replaced by v[p](x[n]-a)<epsilon for every rational epsilon) and the same with the derivative. But every continuous function is picewise-constant. There also exists a non-injective function on Q[p] having the derivative 1 at every point of Q[p] . It should also be noticed that the radius of convergence of the expp(x):=sum(x^n/n!,n=0..infinity) series equals p^(-1) if p >2 and 2^(-2) if p=2. Next, there exists a Haar measure d[p](x)=:dx on Q[p] such that d[p](Z)=1. The definite integral of a real-valued function f(x) over a subset D of Q[p] with respect to dx is defined in certain cases. For example, the definite integral of 1 over

the ball B(0,p^n):={x in Q[p]: v[p](x)<=p^n} with respect to dx equals p^n, ie. the radius of B(0,p^n). It is clear that there does not exist any analog of the Newton-Leibniz formula in the p-adic case. Because of this reason every calculation of every definite p-adic integral is a hard problem.

There are a lot of good and diffent books on p-adic analysis. In particular, see http://www.google.com/search?tbm=bks&tbo=1&q=p-adic&btnG= , http://books.google.com/books?id=H6sq_x2-DgoC&printsec=frontcover&dq=p-adic&hl=uk&ei=IgFuToupO8SL4gTE-tDOBA&sa=X&oi=book_result&ct=result&resnum=6&ved=

0CEYQ6AEwBQ#v=onepage&q&f=false

, and http://books.google.com/books?id=2gTwcJ55QyMC&printsec=frontcover&dq=p-adic&hl=ru&ei=UAxqTuabD5HGtAamhryxBA&sa=

X&oi=book_result&ct=result&resnum=4&ved=0CDkQ6AEwAw#v=onepage&q&f=false as a good introduction to the topic.

Why is it so important? Which are applications? There are indications that the space we live in has not the Archimedean property (see http://en.wikipedia.org/wiki/Archimedean_property) on a very small scale. To verify this hypothesis is a dozen times more expensive than the large hadron collider

(see http://en.wikipedia.org/wiki/Large_Hadron_Collider ). However, the mathematicians already develop the necessary mathematical tools, in particular, p-adic analysis. Concerning other applications, see the answer by Anatoly Kochubei in

http://mathoverflow.net/questions/62866/recent-applications-of-mathematics.

**Edit. **The vanishing text and some typos.