Home » Ordered Fields » Page 2

# Sum of negatives is negative.

Prove that the sum of two negative numbers is negative.

Proof. Let be negative numbers, i.e., and . By Theorem I.25, . Thus, is negative as well.

# Prove some consequences of the order axioms

Prove the following consequences of the order axioms.

1. If and , then .
2. If , then .
3. If , then and are both both positive or and are both negative.
4. If and , then .

1. Proof. First, by definition of we have

Then, by Axiom 8 (and since implies and implies ), we have

Thus, by Axiom 7 , and then using the field properties (Section I.3.3) we have

Hence, , i.e.,

2. Proof. First, implies , and then we have,

Thus, ; hence,

3. Proof. First, we cannot have or since by Theorem I.11 this would mean ; hence, we could not have .
Assume then that and . By assumption . Now, if is positive then

On the other hand, if then

Thus, implies as well. Hence, if is positive then so is , and if is negative then so is , i.e., either and are both positive or both negative

4. Proof. Since we have , and since we have . Then,

Hence,