Why are there opposite rules for dividing positive numbers and negative numbers?

Question

I'm in confusion from some time about division of negative numbers. When we divide a positive number with a positive number, for example $$5/3 = 1.66 $$ we see what is biggest multiple of 3 which is either equal or less than $5$. But in terms of negative numbers, it is not similar to positive numbers. For example, $$ -5/3=-1.66$$ in this example we see the smallest multiple of $3$ which is equal to or greater than $-5$.
My question is why are there two different rules for division?
I'm extremely sorry if I'm asking a foolish question. I'm developing an interest in mathematics and I think I should know the basic thing before proceeding forward. I tried to search about this question but couldn't get results.
Thank you in advance!

Answer 1

Think about it this way: you work with absolute values (so the "biggest" thing applies) and $$+/+ = -/- = +$$ $$+/- = -/+ = -$$ This is what you would find convenient considering $\mathbb{Z}$ as a ring and finding the "smallest number in division sense" (meaning numbers that can't be divided further)... well, plus or minus them.

Answer 2

Probably much more notationally simple to write treat $-1.66$ as $-(1.66)$ than as $(-1)+0.66$

Answer 3

As Thomas Andrews says, remember that multiplication and division have the same operator precedence. That is, the following is true: $$\frac{A\cdot B}{C\cdot D} = (A\cdot B)/(C\cdot D) = A\cdot (B/C)/D = A\cdot (B/(C\cdot D)) = A\cdot (B/D)/C...$$ So what I do when I do division involving negative numbers, is I always factor out the minus sign as a multiplication by negative one ($-1$). Then, I rearrange the terms such that I do the division on the positive numbers, and deal with the $-1$s later. This way, I only remember the positive rule, and the minus sign simply changes the sign of the answer. In your case, I would use the following factorization: $$-5/3 \rightarrow A=-1 ~~ B=5 ~~ C = 1 ~~ D = 3 \rightarrow (A\cdot B)/(C\cdot D) =$$ $$(A\cdot B)/(C\cdot D) = A\cdot (B/D)\cdot 1/C = -1\cdot (5/3)\cdot 1/1=-1\cdot 1.66\cdot 1 = -1.66$$