WOLFRAM|DEMONSTRATIONS PROJECT

p

​
target
2
3
Ï•
base p
2
3
5
7
11
13
17
19
23
29
31
x
0
3
Solution by Newton's method of
2
x
-2 = 0
Suggestions: (p,
x
0
) = (7, 3), (7, 4), (17, 6), (17, 11), (23, 5), (23, 18), (31, 8), (31,23)
n
x
n
x
n
as fraction
x
n
​ as p-adic
check:
2
x
n
as p-adic
0
3
3
3
7
12
7
1
1.83333
11
6
...11111111111111113.0
7
...32065432065432102.0
7
2
1.46212
193
132
...33062113523306213.0
7
...15156400343310002.0
7
3
1.415
72097
50952
...01623525321216213.0
7
...06010335100000002.0
7
4
1.41421
10390190017
7346972688
...02011266421216213.0
7
...10000000000000002.0
7
This Demonstration applies Newton's method to three polynomials with rational coefficients to approximate the irrational numbers
2
,
3
, and
Ï•=(
5
+1)2
(with these decimal approximations to 10 places:
1.414213562
,
1.732050808
, and
1.618033989
). If the iteration begins with a rational number
x
0
, each successive iteration is also rational as shown in column 3, with heading "
x
n
as fraction".
The iteration using Newton’s method for target
t
is
x
n+1
=
2
x
n
+t
2
x
n
.
The iteration using Newton’s method for the golden ratio
Ï•
is
x
n+1
=
2
x
n
+1
2
x
n
-1
.
Column 4 with heading "
x
n
as
p
-adic" and column 5 with heading "check:
2
x
n
as
p
-adic" show very strange looking numbers indeed. These seem to be infinite base
p
numbers. They are in fact numerals, the
p
-adic expansions of
x
n
and
2
x
n
, respectively. In rare cases, suggested by Kurt Hensel in 1897, columns 4 and 5 also converge, in a sense. The "Suggestions" in this Demonstration give the only cases of
p
-adic convergence for
x
n
<p
.
To appreciate how these
p
-adic numbers can represent rational numbers, observe the following. Columns 4 and 5 are like column 2 in that the digits eventually form an infinitely repeating pattern, but to the left (only 17 digits are displayed). Column 4 is like column 3, entitled "
x
n
as fraction" (where
x
n
=
a
n
/
b
n
), in that
a
n
=
b
n
x
n
in base
p
. Similarly, column 5 behaves like
2
x
n
, in that
2
a
n
=
2
b
n
2
x
n
in base
p
.