Q. Negative plus negative equals positive.

Question: Does “negative plus negative equals positive” hold?

1. Step 1 — Clarify notation and meaning

   Write a generic negative number as \(-a\) where \(a>0\). Another negative number is \(-b\) where \(b>0\). The expression “negative plus negative” means adding two numbers of the form \(-a\) and \(-b\).

   What to do: Represent the two negative numbers as \(-a\) and \(-b\), with \(a\) and \(b\) positive.

2. Step 2 — Use the definition of a negative number as a scalar multiple

   Every negative number can be written as \(-1\) times a positive number. So

   \[
   (-a) + (-b) = (-1)\,a + (-1)\,b .
   \]

   What to do: Replace each negative number by \(-1\) times the corresponding positive number.

3. Step 3 — Factor out the common factor using the distributive law

   Apply the distributive property to factor out \(-1\):

   \[
   (-1)\,a + (-1)\,b = (-1)(a + b).
   \]

   What to do: Factor \(-1\) from the sum to obtain a single negative sign times the sum \(a+b\).

4. Step 4 — Reason about the sign of the result

   Since \(a>0\) and \(b>0\), their sum \(a+b\) is positive. Multiplying a positive number by \(-1\) yields a negative number. Therefore

   \[
   (-a) + (-b) = -(a + b) \quad\text{and}\quad -(a + b) < 0. \]

   What to do: Conclude that the sum is the negative of a positive number, hence negative.

5. Step 5 — Numerical example

   Take concrete numbers, for example \(a=3\) and \(b=2\):

   \[
   (-3) + (-2) = -(3 + 2) = -5.
   \]

   What to do: Verify with numbers to see the result is negative, not positive.

6. Step 6 — Number-line interpretation

   Adding a negative number means moving to the left on the number line. Starting at zero, adding two negatives means moving left twice; the final position is left of zero, i.e. negative.

   What to do: Visualize each addition of a negative as a leftward step; two leftward steps cannot land on the positive side.

7. Optional: Proof by contradiction

   Assume for contradiction that \((-a)+(-b)>0\). Then

   \[
   – (a + b) > 0.
   \]

   Multiplying both sides of the inequality by \(-1\) (which reverses the inequality) gives

   \[
   a + b < 0, \]

   which contradicts \(a>0\) and \(b>0\). Thus the assumption is false and the sum must be negative.

   What to do: Use contradiction to confirm the sign must be negative.

Conclusion: The statement “negative plus negative equals positive” is false. For any positive numbers \(a\) and \(b\),

\[
(-a) + (-b) = -(a + b) < 0. \]

.