Thread: reciprocal in ordered field

1. reciprocal in ordered field

Let $\displaystyle \mathbb{F}$ be an ordered field and $\displaystyle a \in \mathbb{F}$. If $\displaystyle a > 0$, show that $\displaystyle a^{-1} > 0$. If $\displaystyle a < 0$, show that $\displaystyle a^{-1} < 0$.

So let $\displaystyle a,b,c \in \mathbb{F}$. If $\displaystyle ab = c$ and any two of $\displaystyle a,b, \ \text{or} \ c$ is positive, then so is the third.

Take $\displaystyle b = a^{-1}$ and the Corollary is shown?

Is this correct? Generally, with corollaries, you invoke the theorem?

2. Originally Posted by particlejohn
Let $\displaystyle \mathbb{F}$ be an ordered field and $\displaystyle a \in \mathbb{F}$. If $\displaystyle a > 0$, show that $\displaystyle a^{-1} > 0$. If $\displaystyle a < 0$, show that $\displaystyle a^{-1} < 0$.

So let $\displaystyle a,b,c \in \mathbb{F}$. If $\displaystyle ab = c$ and any two of $\displaystyle a,b, \ \text{or} \ c$ is positive, then so is the third.

Take $\displaystyle b = a^{-1}$ and the Corollary is shown?

Is this correct? Generally, with corollaries, you invoke the theorem?
Well thats correct (since it should be easy to establish 1 > 0 by your axioms)