# noname

**Abstract**

The p-adic arithmetic allows error-free representation of fractions and
error -free arithmetic

using fractions. In this tutorial, we describe infinite- precision p -adic
arithmetic which is more

suitable for software implementations and finite-precision p-adic arithmetic
which is more suitable

for hardware implementations. The finite-precision p-adic representation is also
called

Hensel code which has certain interesting properties and some open problems for
research.

**1 Introduction**

A p- adic number α can be uniquely written in the form

where each of ∈ [0, p
− 1] and the p-adic norm of the number α is defined as
. Note

that the series

converges to in the p-adic norm. Now, as an example, consider the power series expansion

Since converges to , we have

Taking p = 5, we obtain 5-adic expansion of , which can be written in the form

There is a one-to-one correspondence between the power series expansion

and the short representation
, where only the coefficients of the powers
of p are

shown. We can use the p-adic point as a device for displaying the sign of n .

For example,

**2 Representation of Negative Numbers **

If

then

where and for i > n. Thus, for example,

However, watch for leading zeros , they remain unchanged:

**3 Representation of Integers**

Since a positive integer h can be expressed in exactly one way as the sum of
powers of a prime p,

i.e.,

with ∈ [0, p − 1],
there is essentially no difference between p-adic and p-ary representation of h.

The only difference is that the digits in the p -adic representation are written
in reverse order . For

example,

**4 Representation of Rational Numbers **

If α is a rational number , then it has a repeating pattern of
in its p-adic expansion, i.e., it is
of

the form

For example, , and , etc. Let α have the p-adic expansion

where and p divides neither nor . The p-adic expansion for is

and thus

In other words, we compute by

Next, we use

where and p divides neither nor . The p-adic expansion for is

and so

We continue this process until the period is exhibited. Let α = 2/15 and p = 5. Thus,

and n = -1. The 5-adic expansion of 2/15 is found as

which gives us .

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