I am answering exercise $1.2.4:$
I'm letting the other problems in here just for reference of what was already done so that it's clear what we can use. The answer to $1.2.4$ is:
I am a bit confused: For example, in the first case he comes to the conclusion that $(-a)b>0$, in my mind from here we just need to write $(-a)b=-ab>0$ and then add $ab$ in both sides so that we get $0>ab$ but instead of doing that, he chose to write $ab=[-(-a)b]=(-1)(-a)b=-((-a)b)<0$. I get what he is doing in this expression, I just don't get why he did that instead of what I pointed out.
I am trying to understand the following subtleties here:
- Does this happens because we don't know that $a(-b)=-ab$?
I once saw a proof of this fact that was something like:
$$(b+(-b))=0\tag{$\text{Inverse for +}$}$$
$$a(b+(-b))=0 \tag{$a\cdot0=0$}$$
$$ab+a(-b)=0 \tag{$\text{Distributivity}$}$$
$$ a(-b)=-ab\tag{$\text{Subtract ab}$}$$
- I am a bit confused: Is this proof correct? I just noted that in it, we don't mention any ordering axioms at all which seems to be where the ideas of "positive/negative" come from, although, it seems there exists some "similarity" in the meaning of "positive/negative" and additive inverses.