Q. a negative minus a negative equals

Problem

Interpret the phrase “a negative minus a negative” as the algebraic expression where one negative number is subtracted from another negative number. Using variables, write this as

\[ -a – (-b) \]

1. Understand subtraction as adding the additive inverse.

   For any real numbers x and y, x – y means x + (−y). So rewrite the expression by replacing the subtraction with addition of the additive inverse:

   \[ -a – (-b) = -a + \bigl(-(-b)\bigr) \]

2. Simplify the double negative.

   The negative of a negative number equals the original positive number: for any number y, −(−y) = y. Apply this to the second term:

   \[ -a + \bigl(-(-b)\bigr) = -a + b \]

3. Use commutativity of addition (optional rearrangement).

   Because addition is commutative, you may write the sum in the more familiar subtraction form with the positive term first:

   \[ -a + b = b + (-a) = b – a \]

4. Conclusion (general case).

   Therefore, subtracting a negative is the same as adding the corresponding positive. In symbols:

   \[ -a – (-b) = -a + b = b – a \]

5. Special case: both negatives are the same.

   If the two negatives are the same number (that is, b = a), then

   \[ -a – (-a) = -a + a = 0 \]

6. Numeric example (to illustrate).

   Take a = 3 and b = 5:

   \[ -3 – (-5) = -3 + 5 = 2 \]

Final rule: “a negative minus a negative” becomes “add the positive”: \[ -a – (-b) = -a + b = b – a \]. If the two negatives are identical, the result is 0: \[ -a – (-a) = 0 \].