Decimal representation, Cantor set, p-adic
numbers and Ostrowski Theorem.
A. Aizenbud
Weizmann Institute of Science
http://www.wisdom.weizmann.ac.il/~aizenr
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 1 / 8
Approaches to the real numbers
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
Dedekind sections
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
Dedekind sections
decimal representations
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
Dedekind sections
decimal representations
The last is the most elementary but have some problems:
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
Dedekind sections
decimal representations
The last is the most elementary but have some problems:
Hard to perform simple operations:
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
Dedekind sections
decimal representations
The last is the most elementary but have some problems:
Hard to perform simple operations:
0.666 · · · + 0.33 · · · 33233 · · · = 0. · · ·
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
Dedekind sections
decimal representations
The last is the most elementary but have some problems:
Hard to perform simple operations:
0.666 · · · + 0.33 · · · 33233 · · · = 0. · · ·
but
0.666 · · · + 0.33 · · · 33433 · · · = 1. · · ·
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
Dedekind sections
decimal representations
The last is the most elementary but have some problems:
Hard to perform simple operations:
0.666 · · · + 0.33 · · · 33233 · · · = 0. · · ·
but
0.666 · · · + 0.33 · · · 33433 · · · = 1. · · ·
Multiple representation:
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Approaches to the real numbers
axiomatic approach
Cauchy sequences
Dedekind sections
decimal representations
The last is the most elementary but have some problems:
Hard to perform simple operations:
0.666 · · · + 0.33 · · · 33233 · · · = 0. · · ·
but
0.666 · · · + 0.33 · · · 33433 · · · = 1. · · ·
Multiple representation:
0.9999 · · · = 1.000 · · ·
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 2 / 8
Decimal representation
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 3 / 8
Decimal representation
Decimal representation can be viewed as a map
D : {0, . . . , 9}N → [0, 1].
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 3 / 8
Decimal representation
Decimal representation can be viewed as a map
D : {0, . . . , 9}N → [0, 1].
This map is
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 3 / 8
Decimal representation
Decimal representation can be viewed as a map
D : {0, . . . , 9}N → [0, 1].
This map is
Onto
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 3 / 8
Decimal representation
Decimal representation can be viewed as a map
D : {0, . . . , 9}N → [0, 1].
This map is
Onto
Continuous
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 3 / 8
Decimal representation
Decimal representation can be viewed as a map
D : {0, . . . , 9}N → [0, 1].
This map is
Onto
Continuous
A topological quotient
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 3 / 8
Decimal representation
Decimal representation can be viewed as a map
D : {0, . . . , 9}N → [0, 1].
This map is
Onto
Continuous
A topological quotient
BUT NOT 1-1
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 3 / 8
Decimal representation
Decimal representation can be viewed as a map
D : {0, . . . , 9}N → [0, 1].
This map is
Onto
Continuous
A topological quotient
BUT NOT 1-1
If it would be 1-1 it would be an homeomorphism.
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 3 / 8
The Cantor set
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 4 / 8
The Cantor set
Exercise
{0, . . . , 9}N ∼
=
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 4 / 8
The Cantor set
Exercise
{0, . . . , 9}N ∼
=
= {0, . . . , n}N ∼
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 4 / 8
The Cantor set
Exercise
Q ∼
Fi =
{0, . . . , 9}N ∼
=
= {0, . . . , n}N ∼
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 4 / 8
The Cantor set
Exercise
Q ∼
Fi = {(0.a1 · · · )3 |ai ∈ {0, 1}}
{0, . . . , 9}N ∼
=
= {0, . . . , n}N ∼
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 4 / 8
The Cantor set
Exercise
Q ∼
Fi = {(0.a1 · · · )3 |ai ∈ {0, 1}} ∼
{0, . . . , 9}N ∼
=
=
= {0, . . . , n}N ∼
Cantor set
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 4 / 8
The Cantor set
Exercise
Q ∼
Fi = {(0.a1 · · · )3 |ai ∈ {0, 1}} ∼
{0, . . . , 9}N ∼
=
=
= {0, . . . , n}N ∼
Cantor set
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 4 / 8
p-adic numbers
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 5 / 8
p-adic numbers
Definition
Qp = {(· · · a0 .a1 · · · an )|ai ∈ {0, . . . , p − 1}}
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 5 / 8
p-adic numbers
Definition
Qp = {(· · · a0 .a1 · · · an )|ai ∈ {0, . . . , p − 1}}
+, −, · are easy to define
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 5 / 8
p-adic numbers
Definition
Qp = {(· · · a0 .a1 · · · an )|ai ∈ {0, . . . , p − 1}}
+, −, · are easy to define
Exercise
One can define “:” iff p is prime.
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 5 / 8
p-adic numbers
Definition
Qp = {(· · · a0 .a1 · · · an )|ai ∈ {0, . . . , p − 1}}
+, −, · are easy to define
Exercise
One can define “:” iff p is prime.
Definition (p-adic absolute value)
|pn · ba |p := p−n , where (a, p) = (b, p) = 1
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 5 / 8
p-adic numbers
Definition
Qp = {(· · · a0 .a1 · · · an )|ai ∈ {0, . . . , p − 1}}
+, −, · are easy to define
Exercise
One can define “:” iff p is prime.
Definition (p-adic absolute value)
|pn · ba |p := p−n , where (a, p) = (b, p) = 1
Exercise
Equivalent definition of Qp : Completion of Q w.r.t. | · |p
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 5 / 8
Absolute Value
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
|a + b| ≤ |a| + |b|.
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
|a + b| ≤ |a| + |b|.
| · |α and | · |β are equivalent iff they define the same
topology (i.e. the same notion of convergence)
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
|a + b| ≤ |a| + |b|.
| · |α and | · |β are equivalent iff they define the same
topology (i.e. the same notion of convergence)
Exercise
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
|a + b| ≤ |a| + |b|.
| · |α and | · |β are equivalent iff they define the same
topology (i.e. the same notion of convergence)
Exercise
| · |α and | · |β are equivalent iff | · |aα = | · |β
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
|a + b| ≤ |a| + |b|.
| · |α and | · |β are equivalent iff they define the same
topology (i.e. the same notion of convergence)
Exercise
| · |α and | · |β are equivalent iff | · |aα = | · |β
We have the following dichotomy
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
|a + b| ≤ |a| + |b|.
| · |α and | · |β are equivalent iff they define the same
topology (i.e. the same notion of convergence)
Exercise
| · |α and | · |β are equivalent iff | · |aα = | · |β
We have the following dichotomy
Either | · | is Archimedean, i.e. |Z| is unbounded
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
|a + b| ≤ |a| + |b|.
| · |α and | · |β are equivalent iff they define the same
topology (i.e. the same notion of convergence)
Exercise
| · |α and | · |β are equivalent iff | · |aα = | · |β
We have the following dichotomy
Either | · | is Archimedean, i.e. |Z| is unbounded
Or | · | is non-Archimedean, i.e. |a + b| ≤ max(|a|, |b|).
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Absolute Value
Definition (absolute value)
absolute value on a field F is a map | · | : F → R≥0 s.t.
|a| = 0 iff a=0,
|ab| = |a||b|,
|a + b| ≤ |a| + |b|.
| · |α and | · |β are equivalent iff they define the same
topology (i.e. the same notion of convergence)
Exercise
| · |α and | · |β are equivalent iff | · |aα = | · |β
We have the following dichotomy
Either | · | is Archimedean, i.e. |Z| is unbounded
Or | · | is non-Archimedean, i.e. |a + b| ≤ max(|a|, |b|).
Show that if | · | is non-Archimedean then | · |a is also a
(non-Archimedean) absolute value
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 6 / 8
Ostrowski Theorem
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 7 / 8
Ostrowski Theorem
Theorem (Ostrowski)
the possible (up to equivalence) absolute values on Q are:
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 7 / 8
Ostrowski Theorem
Theorem (Ostrowski)
the possible (up to equivalence) absolute values on Q are:
Archimedean: the usual one – | · |∞ .
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 7 / 8
Ostrowski Theorem
Theorem (Ostrowski)
the possible (up to equivalence) absolute values on Q are:
Archimedean: the usual one – | · |∞ .
non-Archimedean: the p-adic ones – | · |p .
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 7 / 8
Ostrowski Theorem
Theorem (Ostrowski)
the possible (up to equivalence) absolute values on Q are:
Archimedean: the usual one – | · |∞ .
non-Archimedean: the p-adic ones – | · |p .
Exercise
What powers of | · |∞ are absolute value
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 7 / 8
Local Fields
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8
Local Fields
Definition (Local Field)
A local field is a locally compact non-discrete topological field.
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8
Local Fields
Definition (Local Field)
A local field is a locally compact non-discrete topological field.
Theorem (Classification of Local Fields)
The possible local fields are:
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8
Local Fields
Definition (Local Field)
A local field is a locally compact non-discrete topological field.
Theorem (Classification of Local Fields)
The possible local fields are:
positive Characteristic: Fq ((t))
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8
Local Fields
Definition (Local Field)
A local field is a locally compact non-discrete topological field.
Theorem (Classification of Local Fields)
The possible local fields are:
positive Characteristic: Fq ((t))
zero Characteristic:
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8
Local Fields
Definition (Local Field)
A local field is a locally compact non-discrete topological field.
Theorem (Classification of Local Fields)
The possible local fields are:
positive Characteristic: Fq ((t))
zero Characteristic:
non-Archimedean: finite extortions of Qp
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8
Local Fields
Definition (Local Field)
A local field is a locally compact non-discrete topological field.
Theorem (Classification of Local Fields)
The possible local fields are:
positive Characteristic: Fq ((t))
zero Characteristic:
non-Archimedean: finite extortions of Qp
Archimedean:
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8
Local Fields
Definition (Local Field)
A local field is a locally compact non-discrete topological field.
Theorem (Classification of Local Fields)
The possible local fields are:
positive Characteristic: Fq ((t))
zero Characteristic:
non-Archimedean: finite extortions of Qp
Archimedean:
R
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8
Local Fields
Definition (Local Field)
A local field is a locally compact non-discrete topological field.
Theorem (Classification of Local Fields)
The possible local fields are:
positive Characteristic: Fq ((t))
zero Characteristic:
non-Archimedean: finite extortions of Qp
Archimedean:
R
C
A. Aizenbud
Decimal representation, Cantor set and p-adic numbers. 8 / 8