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.

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Stumbling Robot

A Fraction of a Dot
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Tag: Ordered Fields

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Sum of negatives is negative.

* Proof. * Let be negative numbers, i.e., and . By Theorem I.25, . Thus, is negative as well.
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Prove some consequences of the order axioms

Prove that the sum of two negative numbers is negative.

Prove the following consequences of the order axioms.

- If and , then .
- If , then .
- If , then and are both both positive or and are both negative.
- If and , then .

*Proof.*First, by definition of we haveThen, 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.,

*Proof.*First, implies , and then we have,Thus, ; hence,

*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 thenOn 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

*Proof.*Since we have , and since we have . Then,Hence,