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Visual Representations ofp-adic Numbers
Mark Pedigo
Saint Louis University
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Introducing p-adic
numbers(1897) The p-adic numbers were first
introduced by Kurt Hensel.
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Introducing p-adic
numbers(1897) The p-adic numbers were first
introduced by Kurt Hensel.
He used them to bring the methods of powerseries into number theory.
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Introducing p-adic
numbers(1897) The p-adic numbers were first
introduced by Kurt Hensel.
He used them to bring the methods of powerseries into number theory.
p-adic Analysis is now a subject in its ownright.
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The p-adic normGiven q Q, write q = a
bpn for a,b,n Z,
where the prime p divides neither a nor b.
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The p-adic normGiven q Q, write q = a
bpn for a,b,n Z,
where the prime p divides neither a nor b.
p-adic norm
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The p-adic normGiven q Q, write q = a
bpn for a,b,n Z,
where the prime p divides neither a nor b.
p-adic norm
If q = 0, |q|p = |a
b pn
|p =1
pn
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p-adic norm
examplesExamples
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p-adic norm
examplesExamples
|75|5 = |3 52|5 = 152 = 125
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p-adic norm
examplesExamples
|75|5 = |3 52|5 = 152 = 125| 2375|5
= |23 53|
5= 53 = 125
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p-adic norm
examplesExamples
|75|5 = |3 52|5 = 152 = 125| 2375|5
= |23 53|
5= 53 = 125
|3|5 = |4|5 = |7|5 = |12
7 |5 =1
50 = 1
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The p-adic metricBasic idea: Two points are close if their
difference is divisible by a large power of aprime p
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The p-adic metricBasic idea: Two points are close if their
difference is divisible by a large power of aprime p
d(x, y) = |x y|p
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The p-adic metricBasic idea: Two points are close if their
difference is divisible by a large power of aprime p
d(x, y) = |x y|p
Example. 7-adic metric: d(2, 51) < d(1, 2)
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The p-adic metricBasic idea: Two points are close if their
difference is divisible by a large power of aprime 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 =
1
72= 1
49
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The p-adic metricBasic idea: Two points are close if their
difference is divisible by a large power of aprime 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 =
1
72= 1
49
d(1, 2) = |2 1|7 = |1|7 = |70|7 = 170 =11
= 1
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p-adic expansionsp-adic expansionof any q Q:
q =
k=n akpk for some n Z,ak 0, 1, . . . , p 1 for each k n.
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p-adic expansionsp-adic expansionof any q Q:
q =
k=n akpk for some n Z,ak 0, 1, . . . , p 1 for each k n.
We sometimes denote q by its digits; i.e.,q = a1a2a3 . . . ar
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p-adic expansionsp-adic expansionof any q Q:
q =
k=n akpk for some n Z,ak 0, 1, . . . , p 1 for each k n.
We sometimes denote q by its digits; i.e.,q = a1a2a3 . . . ar
This means that the digits are represented
backwards
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Example of a p adic
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Example of a p-adic
expansionWhen p = 5,
23.41= 2 52 + 3 51 + 4 50 + 1 51
= 225
+ 35
+ 4 + 5
= 9 1725
= 24225
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C d h
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Convergence and the
value of -1Claim. Under the 3-adic metric,
1 = .222222...
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Convergence and the
value of -1Claim. Under the 3-adic metric,
1 = .222222...Proof
limn |(2 + 2 3 + 2 32
+ + 2 3n
) (1)|
= limn
|3 + 2 3 + 2 32 + + 2 3n|3
= limn
|3n+1|3
= 0.
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di N b
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p-adic Numbers
DefinitionEvery rational number - expressible as a
p-adic expansion
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p-adic Numbers
DefinitionEvery rational number - expressible as a
p-adic expansionNot every p-adic expansion is a rationalnumber
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p-adic Numbers
DefinitionEvery rational number - expressible as a
p-adic expansionNot every p-adic expansion is a rationalnumber
Qp, the field of p-adic numbers: every p-adicexpansion
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A Tree for Z3Z3 = integers in Q3
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A Tree for Z3Z3 = integers in Q3
A tree representation of Z3
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Sierpinski Triangle
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S3,n: replace each triangular region T with
three smaller triangles
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Generalizing the
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Generalizing the
Sierpinski TriangleS3,n: replace each triangular region T with
three smaller trianglesS3 =
n=1S3,n
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Construction of S3
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Z3 and S3
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Albert A. Cuoco. Visualizing the p-adic integers.
Amer. Math. Monthly, 98:355364, 1991
Fernando Q. Gouvea. p-adic Numbers, An
Introduction, Second Edition. Springer, 1991
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Jan E. Holly. Pictures of ultrametric spaces, the
p-adic numbers, and valued fields. Amer. Math.Monthly, 108(8):721728, 2001
Jan E. Holly. Canonical forms for definablesubsets of algebraically closed and real closedvalued fields. J. Symbolic Logic, 60:843860,
1995
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