Q. Negative plus a negative equals what? \( (-a) + (-b) = -(a+b) \).

Answer

Negative plus a negative equals a negative. More precisely, if you add two negative numbers you get a negative number whose absolute value is the sum of the absolute values of the two addends.

1. Set up the general case.

   Let the two negative numbers be \( -a \) and \( -b \), where \( a>0 \) and \( b>0 \).

2. Rewrite negatives as multiplication by −1.

   Use the fact that a negative number can be written as \( -x = (-1)\cdot x \). So

   \( -a + -b = (-1)\cdot a + (-1)\cdot b \)

3. Factor out the common factor using the distributive property.

   Factor \( (-1) \) from both terms:

   \( (-1)\cdot a + (-1)\cdot b = (-1)\cdot (a + b) \)

4. Interpret the result.

   The expression \( (-1)\cdot (a + b) \) is simply the negative of \( a+b \), so

   \( -a + -b = -(a + b) \)

   This shows the sum is negative, with magnitude equal to the sum of the magnitudes.

5. Concrete example.

   For instance, \( -3 + -5 = -(3 + 5) = -8 \).

Therefore: negative plus a negative equals a negative; mathematically \( -a + -b = -(a+b) \).