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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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C d th

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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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di N b

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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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di N b

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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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