Visual Representations of
p-adic Numbers
Mark Pedigo
Saint Louis University
Visual Representations of p-adic Numbers – p. 1/17
Introducing p-adic
numbers
(1897) The p-adic numbers were first
introduced by Kurt Hensel.
Visual Representations of p-adic Numbers – p. 2/17
Introducing p-adic
numbers
(1897) The p-adic numbers were first
introduced by Kurt Hensel.
He used them to bring the methods of power
series into number theory.
Visual Representations of p-adic Numbers – p. 2/17
Introducing p-adic
numbers
(1897) The p-adic numbers were first
introduced by Kurt Hensel.
He used them to bring the methods of power
series into number theory.
p-adic Analysis is now a subject in its own
right.
Visual Representations of p-adic Numbers – p. 2/17
The p-adic norm
Given q ∈ Q, write q = ab · pn for a, b, n ∈ Z,
where the prime p divides neither a nor b.
Visual Representations of p-adic Numbers – p. 3/17
The p-adic norm
Given q ∈ Q, write q = ab · pn for a, b, n ∈ Z,
where the prime p divides neither a nor b.
p-adic norm
Visual Representations of p-adic Numbers – p. 3/17
The p-adic norm
Given q ∈ Q, write q = ab · pn for a, b, n ∈ Z,
where the prime p divides neither a nor b.
p-adic norm
If q 6= 0, |q|p = | ab · pn |p =
1
pn
Visual Representations of p-adic Numbers – p. 3/17
The p-adic norm
Given q ∈ Q, write q = ab · pn for a, b, n ∈ Z,
where the prime p divides neither a nor b.
p-adic norm
If q 6= 0, |q|p = | ab · pn |p =
1
pn
|0|p = 0
Visual Representations of p-adic Numbers – p. 3/17
p-adic norm
examples
Examples
Visual Representations of p-adic Numbers – p. 4/17
p-adic norm
examples
Examples
|75|5 = |3 · 52 |5 =
1
52
=
1
25
Visual Representations of p-adic Numbers – p. 4/17
p-adic norm
examples
Examples
|75|5 = |3 · 52 |5 =
1
52
=
1
25
2
|5 = | 23 · 5−3 |5 = 53 = 125
| 375
Visual Representations of p-adic Numbers – p. 4/17
p-adic norm
examples
Examples
|75|5 = |3 · 52 |5 =
1
52
=
1
25
2
|5 = | 23 · 5−3 |5 = 53 = 125
| 375
|3|5 = |4|5 = |7|5 = | 12
7 |5 =
1
50
=1
Visual Representations of p-adic Numbers – p. 4/17
The p-adic metric
Basic idea: Two points are “close” if their
difference is divisible by a large power of a
prime p
Visual Representations of p-adic Numbers – p. 5/17
The p-adic metric
Basic idea: Two points are “close” if their
difference is divisible by a large power of a
prime p
d(x, y) = |x − y|p
Visual Representations of p-adic Numbers – p. 5/17
The p-adic metric
Basic idea: Two points are “close” if their
difference is divisible by a large power of a
prime p
d(x, y) = |x − y|p
Example. 7-adic metric: d(2, 51) < d(1, 2)
Visual Representations of p-adic Numbers – p. 5/17
The p-adic metric
Basic idea: Two points are “close” if their
difference is divisible by a large power of a
prime p
d(x, y) = |x − y|p
Example. 7-adic metric: d(2, 51) < d(1, 2)
d(2, 51) = |51 − 2|7 = |49|7 = |72 |7 = 712 =
1
49
Visual Representations of p-adic Numbers – p. 5/17
The p-adic metric
Basic idea: Two points are “close” if their
difference is divisible by a large power of a
prime p
d(x, y) = |x − y|p
Example. 7-adic metric: d(2, 51) < d(1, 2)
d(2, 51) = |51 − 2|7 = |49|7 = |72 |7 = 712 =
d(1, 2) = |2 − 1|7 = |1|7 = |70 |7 =
1
70
=
1
1
1
49
=1
Visual Representations of p-adic Numbers – p. 5/17
p-adic expansions
p-adic
expansion
of
any
q
∈
Q:
P∞
q = k=n ak pk for some n ∈ Z,
ak ∈ 0, 1, . . . , p − 1 for each k ≥ n.
Visual Representations of p-adic Numbers – p. 6/17
p-adic expansions
p-adic
expansion
of
any
q
∈
Q:
P∞
q = k=n ak pk for some n ∈ Z,
ak ∈ 0, 1, . . . , p − 1 for each k ≥ n.
We sometimes denote q by its digits; i.e.,
q = a1 a2 a3 . . . ar
Visual Representations of p-adic Numbers – p. 6/17
p-adic expansions
p-adic
expansion
of
any
q
∈
Q:
P∞
q = k=n ak pk for some n ∈ Z,
ak ∈ 0, 1, . . . , p − 1 for each k ≥ n.
We sometimes denote q by its digits; i.e.,
q = a1 a2 a3 . . . ar
This means that the digits are represented
“backwards”
Visual Representations of p-adic Numbers – p. 6/17
Example of a p-adic
expansion
When p = 5,
23.41
= 2 · 5−2 + 3 · 5−1 + 4 · 50 + 1 · 51
2
= 25
+ 53 + 4 + 5
= 9 17
25
= 242
25
Visual Representations of p-adic Numbers – p. 7/17
Convergence and the
value of -1
Claim. Under the 3-adic metric,
−1 = .222222...
Visual Representations of p-adic Numbers – p. 8/17
Convergence and the
value of -1
Claim. Under the 3-adic metric,
−1 = .222222...
Proof
lim |(2 + 2 · 3 + 2 · 32 + · · · + 2 · 3n ) − (−1)|3
n→∞
= lim |3 + 2 · 3 + 2 · 32 + · · · + 2 · 3n |3
n→∞
= lim |3n+1 |3
n→∞
= 0.
Visual Representations of p-adic Numbers – p. 8/17
p-adic Numbers
Definition
Every rational number - expressible as a
p-adic expansion
Visual Representations of p-adic Numbers – p. 9/17
p-adic Numbers
Definition
Every rational number - expressible as a
p-adic expansion
Not every p-adic expansion is a rational
number
Visual Representations of p-adic Numbers – p. 9/17
p-adic Numbers
Definition
Every rational number - expressible as a
p-adic expansion
Not every p-adic expansion is a rational
number
Qp , the field of p-adic numbers: every p-adic
expansion
Visual Representations of p-adic Numbers – p. 9/17
A Tree for Z3
Z3 = integers in Q3
Visual Representations of p-adic Numbers – p. 10/17
A Tree for Z3
Z3 = integers in Q3
A tree representation of Z3
Visual Representations of p-adic Numbers – p. 10/17
A Tree Path for -1
Figure 1: The path of −1 = .22222... in the tree
representation of Z3
Visual Representations of p-adic Numbers – p. 11/17
Sierpinski Triangle
Visual Representations of p-adic Numbers – p. 12/17
S3,n : replace each triangular region T with
three smaller triangles
Visual Representations of p-adic Numbers – p. 13/17
Generalizing the
Sierpinski Triangle
S3,n : replace each triangular region T with
three smaller triangles
S3 = ∪∞
n=1 S3,n
Visual Representations of p-adic Numbers – p. 13/17
Construction of S3
Visual Representations of p-adic Numbers – p. 14/17
Z3 and S3
Visual Representations of p-adic Numbers – p. 15/17
Albert A. Cuoco. Visualizing the p-adic integers.
Amer. Math. Monthly, 98:355–364, 1991
Fernando Q. Gouvea. p-adic Numbers, An
Introduction, Second Edition. Springer, 1991
Visual Representations of p-adic Numbers – p. 16/17
Jan E. Holly. Pictures of ultrametric spaces, the
p-adic numbers, and valued fields. Amer. Math.
Monthly, 108(8):721–728, 2001
Jan E. Holly. Canonical forms for definable
subsets of algebraically closed and real closed
valued fields. J. Symbolic Logic, 60:843–860,
1995
Visual Representations of p-adic Numbers – p. 17/17