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1.3 The real numbers  (Page 2/3)

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An ordered field satisfies the familiar laws of inequalities. They are consequences of the two properties of the set P .

Using the positivity properties above for an ordered field F , together with the axioms for a field, derive the familiar laws of inequalities:

  1. (Transitivity) If x < y and y < z , then x < z .
  2. (Adding like inequalities) If x < y and z < w , then x + z < y + w .
  3. If x < y and a > 0 , then a x < a y .
  4. If x < y and a < 0 , then a y < a x .
  5. If 0 < a < b and 0 < c < d , then a c < b d .
  6. Verify parts (a) through (e) with < replaced by ≤ .
  7. If x and y are elements of F , show that one and only one of the following three relations can hold: (i) x < y , (ii) x > y , (iii) x = y .
  8. Suppose x and y are elements of F , and assume that x ≤ y and y ≤ x . Prove that x = y .
  1. If F is an ordered field, show that 1 ∈ P ; i.e., that 0 < 1 . HINT: By the law of tricotomy, only one of the three possibilities holds for 1 . Rule out the last two.
  2. Show that F 7 of [link] is not an ordered field; i.e., there is no subset P ⊆ F 7 such that the two positivity properties can hold. HINT: Use part (a) and positivity property (1).
  3. Prove that Q is an ordered field, where the set P is taken to be the usual set of positive rational numbers. That is, P consists of those rational numbers a / b for which both a and b are natural numbers.
  4. Suppose F is an ordered field and that x is a nonzero element of F . Show that for all natural numbers n n x ≠ 0 .
  5. (e) Show that, in an ordered field, every nonzero square is positive; i.e., if x ≠ 0 , then x 2 ∈ P .

We remarked earlier that there are many different examples of fields, and many of these are also ordered fields.Some fields, though technically different from each other, are really indistinguishable from the algebraic point of view, andwe make this mathematically precise with the following definition.

Let F 1 and F 2 be two ordered fields, and write P 1 and P 2 for the set of positive elements in F 1 and F 2 respectively. A 1-1 correspondence J between F 1 and F 2 is called an isomorphism if

  1. J ( x + y ) = J ( x ) + J ( y ) for all x , y ∈ F 1 .
  2. J ( x y ) = J ( x ) J ( y ) for all x , y ∈ F 1 .
  3. x ∈ P 1 if and only if J ( x ) ∈ P 2 .

REMARK. In general, if A 1 and A 2 are two algebraic systems, then a 1-1 correspondence between A 1 and A 2 is called an isomorphism if it converts the algebraic structure on A 1 into the corresponding algebraic structure on A 2 .

  1. Let F be an ordered field. Define a function J : N → F by J ( n ) = n · 1 . Prove that J is an isomorphism of N onto a subset N ˜ of F . That is, show that this correspondence is one-to-one and converts addition and multiplicationin N into addition and multiplication in F . Give an example to show that this result is not true if F is merely a field and not an ordered field.
  2. Let F be an ordered field. Define a function J : Q → F by J ( k / n ) = k · 1 × ( n · 1 ) - 1 . Prove that J is an isomorphism of the ordered field Q onto a subset Q ˜ of the ordered field F . Conclude that every ordered field F contains a subset that is isomorphic to the ordered field Q .

REMARK. Part (b) of [link] shows that the ordered field Q is the smallest possible ordered field, in the sense that every other ordered field contains an isomorphic copy of Q . However, as mentioned earlier, the ordered field Q cannot suffice as the set of real numbers. There is no rational number whose square is 2, and we want the square root of 2 to be a real number. See [link] below. What extra property is there about an ordered field F that will allow us to prove that numbers like 2 , π , and so on are elements of F ? It turns out that the extra property we need is related to a quite subtle point concerning upper and lower bounds of sets. It gives us some initial indication that the known-to-be subtle concept of a limit may be fundamental to our very notion of what the real numbers are.
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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