Slide Ve So Padic
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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
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
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