Early Question on Negative Number

In 17th century, the ratio of positive integers and negative integers are as follows:

-n / n = -1 = n /-n

Above equation seems normal. But we know -n is less than n, and give an equal sign in the expression on the right and the left, ie, dividing the smaller number with larger numbers is equal to dividing the larger number with smaller numbers, which of course we would think it impossible. This is called the paradox of Arnauld.

To accomplish this, we can not assume-n (negative integer) as usual quantity, but as the quantity with a different direction. It looks very natural given the integer applications in everyday life, such as stepping forward and stepping backward, or the money we spend and what we get. With this, we no longer consider the comparison '-n / n' and 'n /-n' as the ratio of quantity only. When the ratio of expanded notation for negative numbers as in the positive numbers, terminology b/c can be understood as the number needed to be multiplied by c to get b. Multiplication number k.-j with negative numbers can be understood as a multiplication with the absolute value but pay attention to the direction of the inverse multiplication k.j.