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].
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].
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 .