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Abstract
The padic arithmetic allows errorfree representation of fractions and
errorfree arithmetic
using fractions . In this tutorial, we describe infinite precision p adic
arithmetic which is more
suitable for software implementations and finiteprecision padic arithmetic
which is more suitable
for hardware implementations. The finiteprecision padic 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 padic norm of the number α is defined as
. Note
that the series
converges to in the padic norm. Now, as an example, consider the power series expansion
Since converges to , we have
Taking p = 5, we obtain 5adic expansion of , which can be written in the form
There is a onetoone 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 padic 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 padic and pary 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 padic expansion, i.e., it is
of
the form
For example, , and , etc. Let α have the padic expansion
where and p divides neither nor . The padic expansion for is
and thus
In other words, we compute by
Next, we use
where and p divides neither nor . The padic 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 5adic expansion of 2/15 is found as
which gives us .
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