Assume { F, +,*,, <} is an ordered field. If a<b in F and c=(a+b)/2, prove

a) a<c<b

b) b - c = c - a = (b-a)/2. Thus the distance from c to either a or b is 1/2 the distance from a to b. C is called the mid point of [a,b].

Printable View

- September 9th 2009, 06:25 PMtigergirlQuestion about ordered field...PLEASE HELP!!!
Assume { F, +,*,, <} is an ordered field. If a<b in F and c=(a+b)/2, prove

a) a<c<b

b) b - c = c - a = (b-a)/2. Thus the distance from c to either a or b is 1/2 the distance from a to b. C is called the mid point of [a,b]. - September 10th 2009, 02:48 AMTaluivren
Hi,

I don't know what your favourite definition of an ordered field is so i'll use this one: A field is an*ordered field*if its nonzero elements can be split into two sets, and , such that contains iff contains , and is closed under addition and multiplication.

The ordering can be then defined as follows: .

so either or , but if then (since is closed under multiplication) which is a contradiction, so and .

Since is closed under addition, .

Furthermore, , because if then and since we'd get (since is closed under multiplication), a contradiction.

Now we're ready to prove a), b).

.

Similarly, .

Since was given and we know , we get . Thus and .