# p

p

This Demonstration applies Newton's method to three polynomials with rational coefficients to approximate the irrational numbers , , and (with these decimal approximations to 10 places: , , and ). If the iteration begins with a rational number , each successive iteration is also rational as shown in column 3, with heading " as fraction".

2

3

ϕ=(

5

+1)21.414213562

1.732050808

1.618033989

x

0

x

n

The iteration using Newton’s method for target is =+t.

t

x

n+1

2

x

n

2

x

n

The iteration using Newton’s method for the golden ratio is =+1.

ϕ

x

n+1

2

x

n

2-1

x

n

Column 4 with heading " as -adic" and column 5 with heading "check: as -adic" show very strange looking numbers indeed. These seem to be infinite base numbers. They are in fact numerals, the -adic expansions of and , 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 -adic convergence for <p.

x

n

p

2

x

n

p

p

p

x

n

2

x

n

p

x

n

To appreciate how these -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 " as fraction" (where =/), in that = in base . Similarly, column 5 behaves like , in that = in base .

p

x

n

x

n

a

n

b

n

a

n

b

n

x

n

p

2

x

n

2

a

n

2

b

n

2

x

n

p