Ordered Fields Bernd Schr oder logo1 Bernd Schr oder Louisiana - PowerPoint PPT Presentation

Positive Rational Numbers Ordered Fields Ordered Fields Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. 1. For all x , y ∈ P, we have x + y ∈ P and xy ∈ P, logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. 1. For all x , y ∈ P, we have x + y ∈ P and xy ∈ P, 2. For all x ∈ Q , exactly one of the following three properties holds. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. 1. For all x , y ∈ P, we have x + y ∈ P and xy ∈ P, 2. For all x ∈ Q , exactly one of the following three properties holds. Either x ∈ P logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. 1. For all x , y ∈ P, we have x + y ∈ P and xy ∈ P, 2. For all x ∈ Q , exactly one of the following three properties holds. Either x ∈ P or − x ∈ P logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. 1. For all x , y ∈ P, we have x + y ∈ P and xy ∈ P, 2. For all x ∈ Q , exactly one of the following three properties holds. Either x ∈ P or − x ∈ P or x = 0 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. 1. For all x , y ∈ P, we have x + y ∈ P and xy ∈ P, 2. For all x ∈ Q , exactly one of the following three properties holds. Either x ∈ P or − x ∈ P or x = 0 . Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. 1. For all x , y ∈ P, we have x + y ∈ P and xy ∈ P, 2. For all x ∈ Q , exactly one of the following three properties holds. Either x ∈ P or − x ∈ P or x = 0 . Proof. Good exercise. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields � � Theorem. The subset P : = [( n , d )] : n ∈ N , d ∈ N ⊆ Q of the rational numbers is called the set of positive rational numbers . It has the following properties. 1. For all x , y ∈ P, we have x + y ∈ P and xy ∈ P, 2. For all x ∈ Q , exactly one of the following three properties holds. Either x ∈ P or − x ∈ P or x = 0 . Proof. Good exercise. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff there is a subset F + ⊆ F logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff there is a subset F + ⊆ F so that 1. For all x , y ∈ F + we have x + y ∈ F + and xy ∈ F + , and logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff there is a subset F + ⊆ F so that 1. For all x , y ∈ F + we have x + y ∈ F + and xy ∈ F + , and 2. 0 ∈ F + , and logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff there is a subset F + ⊆ F so that 1. For all x , y ∈ F + we have x + y ∈ F + and xy ∈ F + , and 2. 0 ∈ F + , and 3. For all x ∈ F \{ 0 } , either x ∈ F + or − x ∈ F + holds. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff there is a subset F + ⊆ F so that 1. For all x , y ∈ F + we have x + y ∈ F + and xy ∈ F + , and 2. 0 ∈ F + , and 3. For all x ∈ F \{ 0 } , either x ∈ F + or − x ∈ F + holds. The subset F + is also called the positive cone of the totally ordered field. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff there is a subset F + ⊆ F so that 1. For all x , y ∈ F + we have x + y ∈ F + and xy ∈ F + , and 2. 0 ∈ F + , and 3. For all x ∈ F \{ 0 } , either x ∈ F + or − x ∈ F + holds. The subset F + is also called the positive cone of the totally ordered field. The elements of F + are called nonnegative logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff there is a subset F + ⊆ F so that 1. For all x , y ∈ F + we have x + y ∈ F + and xy ∈ F + , and 2. 0 ∈ F + , and 3. For all x ∈ F \{ 0 } , either x ∈ F + or − x ∈ F + holds. The subset F + is also called the positive cone of the totally ordered field. The elements of F + are called nonnegative , the elements of F + \{ 0 } are called positive logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields Definition. A field ( F , + , · ) is called a totally ordered field iff there is a subset F + ⊆ F so that 1. For all x , y ∈ F + we have x + y ∈ F + and xy ∈ F + , and 2. 0 ∈ F + , and 3. For all x ∈ F \{ 0 } , either x ∈ F + or − x ∈ F + holds. The subset F + is also called the positive cone of the totally ordered field. The elements of F + are called nonnegative , the elements of F + \{ 0 } are called positive , and the elements x ∈ F \{ 0 } so that − x ∈ F + are called negative . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields What do we already know about totally ordered fields in general and about Q in particular? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields

Positive Rational Numbers Ordered Fields What do we already know about totally ordered fields in general and about Q in particular? 1. There is an order relation x ≤ y iff y − x ∈ F + . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Ordered Fields