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hi there it's Gabrielle and I will be presenting the concept of an ordered field as it applies to the real number system what is an ordered field especially as it applies to the real number system well in order to describe an ordered field it is necessary to first describe what a field is and what an ordered relation is a field as we know is a set with two binary operations of addition and multiplication these operations satisfy the following axioms all of these axioms here can be found in chapter 1 of our textbook however this is only some of them before I get to the definition of an ordered field I will briefly address order relations because what we're going to be looking at here are binary operations an order relation is a set of ordered elements of a set F that satisfy axioms of order one of these axioms is the trichotomy law which is for any st that is an element of F exactly one of the following hold that s equals T that s is less than T or that T is less than s and the other axiom is the transitive law which is if s is less than T and T is less than B then s is less than V now that we can see how these inequalities work we will look at my first proof which is the following let a be XY the elements of the reals suppose that a is less than or equal to X which is less than or equal to B and suppose that a is less than or equal to Y which is less than or equal to V prove that the absolute value of x minus y is less than or equal to B minus a with this proof we can use lemma two point three point nine point four which allows us to use two inequalities in the drawing we see that we have two equality's which are x is less than or equal to b and y is less than or equal to a if we analyze X minus y we see that X is less than or equal to B and as a result 0 is less than or equal to B minus X and Y is greater than or equal to a and as a result 0 is less than or equal to Y minus a and as we can see through algebra X minus y is less than or equal to B minus a we can do the same thing for y minus X and find that Y minus x is less than or equal to 3 minus a therefore by lemma 2 point 3 point 9.4 it has been proven that the absolute value of X minus y is less than or equal to B minus a these principles lead to the rest of the axioms for binary operations which are what you see here and these axioms that you see in red represent an ordered field I will go through them really quickly trichotomy law the transitive law addition law for order multiplication law for order and the non-trivial 'ti law I just want to point out something here that these properties do not characterize real numbers because rational numbers are also an ordered field so before I move on to talk about the one see that distinguishes the reels from all other ordered fields I want to prove lemma two point three point nine point six which is the triangle inequality to show how these ordered relations work before I do the proof I will outline the goal for the proof the goal to prove it is to show that the absolute value of a plus the absolute value of b squared is greater than or equal to the absolute value of a plus b squared i will use the fact that for all X in the set of real numbers that the square of the absolute value of x equals x squared it's a postulate we know for all real numbers a B we will look at the absolute value of a plus the absolute value of B quantity squared equals the absolute value of a squared plus two times the absolute value of a time's the absolute value of B plus the absolute value of B squared if we look at the idea that the absolute value of a is greater than or equal to a as well as the absolute value of B is greater than or equal to B and then incorporate our known that the square root of the absolute value of x equals x squared then we can conclude that this equation is greater than a squared plus two a B plus B squared thus as a result this equals a plus B squared which then equals the absolute value of a plus B quantity squared so it is shown that the absolute value of a plus B squared is less than or equal to a plus the absolute value of B quantity squared as a result it is proven that the absolute value of a plus B is less than or equal to the absolute value of a plus the absolute value of B I'm now going to talk about the one axiom that distinguishes the reals from all other ordered fields in a book it is axiom two point two point four the axiom for the real numbers that states there exists an ordered field that satisfies the least upper bound property I will explain in defining the least upper bound let's address set a let's let R be an ordered field then a set a a subset of the ordered field R with this in mind then set a is bounded below if there exists a number little B that is an element of the reals wearing little a is less than little B for all elements little a that is an element and set a in this case the number B is called the upper bound for a likewise the set a is bounded below if there is a lower bound little C that is an element of R that satisfies the axiom little C is less than or equal to little a for every little a that is an element of a we can refer to the least upper bound as s equals Lu ba ba but I prefer to refer to it as s equals s upa for supreme 'm for the least upper bound we can define the greatest lower bound for the set a in the same way and it can be referred to as I equals GLBA or I equals bi an FA for infimum a so as you can see the supremum is greater than or equal to B for all upper bounds of a also the infimum is less than or equal to little C for all lower bounds of a so now that we have defined the least upper bound and the greatest lower bound let's do another proof I want to do a proof with regards to the least upper bound property as well as one for the greatest lower bound so I want to prove that if little X is an element of the reals and is an upper bound for a and little X is an element of a then there is a least upper bound for a which is little x equals least upper bound a here we see that a is a non empty subset in the reals so for my proof let little X an element in the reals be another upper bound for a because little B is an element of a then little X is less than or equal to B where B is an upper bound for a thus X is less than or equal to all the other upper bound for a therefore x equals the least upper bound for a my second proof here is to show that if little Y an element in the reals is lower bound of a then little Y is an element of a and then a has a greatest lower bound and little y equals the greatest lower bound of a so for this proof let little Y an element in the reals be another lower bound for a because little seasoned element of a and then little Y is greater than or equal to little C where little C is a lower bound for a thus little Y is greater than or equal to all the other lower bounds for a therefore the y equals the greatest lower bound for a thus proven we have just seen how a field has ordered relations and how its bounded fields with relations so what is an ordered field as it relates to real numbers it is a field with the relation and this relation revolves around the least upper bound property the supremo so this all may seem interesting and it's all fascinating but why are we even learning this well because as our author Ethan Bloch states virtually all the major theorems in this text concerning such topics as continuous functions derivatives integrals sequences and series rely upon the least upper bound property consequentially this leads up to the concept of completeness let's assume that there is an ordered field R which hold a group of non-empty subsets one of which is a so if a has an upper bound and that upper bound has a least upper bound a supreme 'm in r then that ordered field is complete this is really important because this property distinguishes the reals from the rationals and with the ordered field property it characterizes the reals hence the set of real numbers is the only set which is a completed ordered field