[Math] Logic behind dividing negative numbers

arithmeticfractions

I've learnt in school that a positive number, when divided by a negative number, and vice-versa, will give us a negative number as a result.On the other hand, a negative number divided by a negative number will give us a positive number.

Given the following equations:

  1. $\frac{-18}{2} = -9$
  2. $\frac{18}{-2} = -9$
  3. $\frac{-18}{-2} = 9$

This is how I would think the logic behind the equation would be:

  1. If I have a debt of 18 dollars , owed to 2 people equally, I would owe each of them $9

    • $\frac{-18}2 = -9$

  2. If I have 18 dollars, owed to 2 people equally, I would thus give them $9 each

    • $\frac{18}{-2} = -9$

However, I can't seem to come to terms with a negative number dividing by a negative number giving me a positive number as a result. What would be the logic behind it?

Also, I think that I have the logic/reasoning for the 2nd example wrong, as it is exactly the same as the reasoning for the first example?Could someone give me a better example of the logic behind the 2nd example?

I would really appreciate it if anyone could enlighten me.

Best Answer

If you have a debt of \$100, and that debt is paid in \$20 increments, you have:

$$\frac{-\$ 100}{-\$20} = 5\ \textrm{payments}.$$

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