Arithmetic With Rational Numbers
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Defining Rational Numbers
Addition and Multiplication
Integers
Arithmetic With Rational Numbers
Bernd Schröder
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition. The relation (a, b) ∼ (c, d) defined by
a·d = b·c
is an equivalence relation on the set Z × Z \ {0} .
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition. The relation (a, b) ∼ (c, d) defined by
a·d = b·c
is an equivalence relation on the set Z × Z \ {0} . The equivalence class of (a, b) under ∼ will be denoted (a, b) .
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition. The relation (a, b) ∼ (c, d) defined by
a·d = b·c
is an equivalence relation on the set Z × Z \ {0} . The equivalence class of (a, b) under ∼ will be denoted (a, b) .
Proof.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition. The relation (a, b) ∼ (c, d) defined by
a·d = b·c
is an equivalence relation on the set Z × Z \ {0} . The equivalence class of (a, b) under ∼ will be denoted (a, b) .
Proof. Exercise.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition. The relation (a, b) ∼ (c, d) defined by
a·d = b·c
is an equivalence relation on the set Z × Z \ {0} . The equivalence class of (a, b) under ∼ will be denoted (a, b) .
Proof. Exercise.
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Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Discussion
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Discussion
1. Definition of ∼ is motivated by
Bernd Schröder
Arithmetic With Rational Numbers
Integers
a c
= iff ad = bc.
b d
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Discussion
a c
1. Definition of ∼ is motivated by = iff ad = bc.
b d
2. 0 must be excluded from the second component
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Discussion
a c
1. Definition of ∼ is motivated by = iff ad = bc.
b d
2. 0 must be excluded from the second component: The
element (0, 0) would be equivalent to all other elements.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Discussion
a c
1. Definition of ∼ is motivated by = iff ad = bc.
b d
2. 0 must be excluded from the second component: The
element (0, 0) would be equivalent to all other elements.
Just excluding (0, 0) is not enough, because addition of
elements (a, 0) and (b, 0) would re-introduce it.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Discussion
a c
1. Definition of ∼ is motivated by = iff ad = bc.
b d
2. 0 must be excluded from the second component: The
element (0, 0) would be equivalent to all other elements.
Just excluding (0, 0) is not enough, because addition of
elements (a, 0) and (b, 0) would re-introduce it. (Or we
would end up with an extra element that absorbs
everything else in addition and multiplication.)
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
Proof (multiplication only).
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) .
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d.
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d. Therefore
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d. Therefore
acb0 d0
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d. Therefore
acb0 d0 = ab0 cd0
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d. Therefore
acb0 d0 = ab0 cd0
= a0 bc0 d
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d. Therefore
acb0 d0 = ab0 cd0
= a0 bc0 d
= a0 c0 bd,
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d. Therefore
acb0 d0 = ab0 cd0
= a0 bc0 d
= a0 c0 bd,
0 0 0 0 which implies that (ac, bd) = (a c , b d )
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d. Therefore
acb0 d0 = ab0 cd0
= a0 bc0 d
= a0 c0 bd,
0 0 0 0 bd) = (a c, b d ) , that is,
which
that
implies
(ac,
0
(a, b) · (c, d) = (a , b0 ) · (c0 , d0 ) .
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Proposition.
The operations
(a, b) + (c, d)
:= (ad + bc,
bd) and
(a, b) · (c, d) := (ac, bd) are well-defined.
0 0 Proof
(multiplication
only).
Let
(a,
b)
= (a , b ) and let
(c, d) = (c0 , d0 ) . Then ab0 = a0 b and cd0 = c0 d. Therefore
acb0 d0 = ab0 cd0
= a0 bc0 d
= a0 c0 bd,
0 0 0 0 bd) = (a c, b d ) , that is,
which
that
implies
(ac,
0
(a, b) · (c, d) = (a , b0 ) · (c0 , d0 ) .
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Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Definition.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Definition.
The
rational numbers Q are the
set of equivalence
classes (a, b) of elements of Z × Z \ {0} under the
equivalence relation ∼.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Definition.
The
rational numbers Q are the
set of equivalence
classes (a, b) of elements of Z × Z \ {0} under the
equivalence
relation
∼.
Addition is defined by
(a, b) + (c, d) := (ad + bc, bd)
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Definition.
The
rational numbers Q are the
set of equivalence
classes (a, b) of elements of Z × Z \ {0} under the
equivalence
relation
∼.
Addition is defined by
(a, b) + (c, d) := (ad + bc,bd) and multiplication is
defined by (a, b) · (c, d) := (ac, bd) .
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
commutative
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
commutative, has a neutral element 0 := (0, 1)
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Multiplication of rational numbers is associative
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Multiplication of rational numbers is associative, commutative
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Multiplication of rational numbers
is associative, commutative,
has a neutral element 1 := (1, 1)
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Multiplication of rational numbers
commutative,
is associative,
,
for
every
(a,
b)
∈ Q \ {0}
has a neutral element
1
:=
(1,
1)
the element (b, a) is its multiplicative inverse
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Multiplication of rational numbers
commutative,
is associative,
,
for
every
(a,
b)
∈ Q \ {0}
has a neutral element
1
:=
(1,
1)
the element (b, a) is its multiplicative inverse, and
multiplication is distributive over addition.
Bernd Schröder
Arithmetic With Rational Numbers
logo1
Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Multiplication of rational numbers
commutative,
is associative,
,
for
every
(a,
b)
∈ Q \ {0}
has a neutral element
1
:=
(1,
1)
the element (b, a) is its multiplicative inverse, and
multiplication is distributive over addition.
Proof.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Multiplication of rational numbers
commutative,
is associative,
,
for
every
(a,
b)
∈ Q \ {0}
has a neutral element
1
:=
(1,
1)
the element (b, a) is its multiplicative inverse, and
multiplication is distributive over addition.
Proof. Exercise.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. Addition of rational numbers is associative,
has a neutral
commutative,
element
0 := (0, 1) and for every
(a, b) ∈ Q the element (−a, b) is its additive inverse.
Multiplication of rational numbers
commutative,
is associative,
,
for
every
(a,
b)
∈ Q \ {0}
has a neutral element
1
:=
(1,
1)
the element (b, a) is its multiplicative inverse, and
multiplication is distributive over addition.
Proof. Exercise.
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Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The set [(a, 1)] : a ∈ Z ⊆ Q is ring isomorphic to
Z.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The set [(a, 1)] : a ∈ Z ⊆ Q is ring isomorphic to
Z.
Proof.
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Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The set [(a, 1)] : a ∈ Z ⊆ Q is ring isomorphic to
Z.
Proof. The
f : Z → [(a, 1)] : a ∈ Z ⊆ Q defined by
function
f (a) = (a, 1) is an isomorphism.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The set [(a, 1)] : a ∈ Z ⊆ Q is ring isomorphic to
Z.
Proof. The
f : Z → [(a, 1)] : a ∈ Z ⊆ Q defined by
function
f (a) = (a, 1) is an isomorphism. (Exercise.)
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The set [(a, 1)] : a ∈ Z ⊆ Q is ring isomorphic to
Z.
Proof. The
f : Z → [(a, 1)] : a ∈ Z ⊆ Q defined by
function
f (a) = (a, 1) is an isomorphism. (Exercise.)
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The set [(a, 1)] : a ∈ Z ⊆ Q is ring isomorphic to
Z.
Proof. The
f : Z → [(a, 1)] : a ∈ Z ⊆ Q defined by
function
f (a) = (a, 1) is an isomorphism. (Exercise.)
Definition.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The set [(a, 1)] : a ∈ Z ⊆ Q is ring isomorphic to
Z.
Proof. The
f : Z → [(a, 1)] : a ∈ Z ⊆ Q defined by
function
f (a) = (a, 1) is an isomorphism. (Exercise.)
Definition. The set [(a, 1)] : a ∈ Z ⊆ Q will also be called
the set of integers Z.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The set [(a, 1)] : a ∈ Z ⊆ Q is ring isomorphic to
Z.
Proof. The
f : Z → [(a, 1)] : a ∈ Z ⊆ Q defined by
function
f (a) = (a, 1) is an isomorphism. (Exercise.)
Definition. The set [(a, 1)] : a ∈ Z ⊆Q will also be called
the set of integers Z. Similarly, the set [(a, 1)] : a ∈ N ⊆ Q
will also be called the set of natural numbers N.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The rational numbers Q are countable.
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The rational numbers Q are countable.
Proof.
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Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The rational numbers Q are countable.
Proof. The set Q =
[
[(n, d)] : d ∈ Z \ {0} is countable.
n∈Z
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The rational numbers Q are countable.
Proof. The set Q =
[
[(n, d)] : d ∈ Z \ {0} is countable.
n∈Z
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
Defining Rational Numbers
Addition and Multiplication
Integers
Theorem. The rational numbers Q are countable.
Proof. The set Q =
[
[(n, d)] : d ∈ Z \ {0} is countable.
n∈Z
Bernd Schröder
Arithmetic With Rational Numbers
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Louisiana Tech University, College of Engineering and Science
