Proving the basic calculation rules for estimates

Question

I always forget the proof why for example a negative number times a positive number is negative again. And the book is not very reader-friendly, i.e. it there is not a lot of prose which I would prefer.

Answer 1

We'll prove that $$(-a)b=-ab.$$ Indeed, $$(-a)b+ab=(-a+a)b=0b$$ and it's enough to prove that $$0b=0,$$ which is true because $$0b=(0+0)b=0b+0b$$ and from here $$0b=0b+(-0b)=0.$$ Now, take $a>0$ and $b>0$ and you'll get the rule.