Q. a positive plus a negative equals

Problem

What does a positive number plus a negative number equal?

Step-by-step explanation

1. Rewrite the numbers with clear signs.

   Let the positive number be \(a\) and the negative number be \(-b\), where \(a > 0\) and \(b > 0\). The expression becomes

   \(a + (-b)\)

2. Recognize that adding a negative is subtraction.

   By the definition of subtraction, adding the negative number is the same as subtracting the positive amount \(b\). So

   \(a + (-b) = a – b\)

3. Compare the magnitudes \(a\) and \(b\) and determine the sign of the result.

   There are three cases to consider:

   1. If \(a > b\):

      The difference is positive. The result is the positive number \(a\) minus \(b\):

      \(a + (-b) = a – b\) which is greater than 0.

      Example: \(7 + (-3) = 7 – 3 = 4\).

   2. If \(a < b\):

      The difference is negative. You can write the result as the negative of the difference of magnitudes:

      \(a + (-b) = -(b – a)\).

      Example: \(3 + (-7) = 3 – 7 = -(7 – 3) = -4\).

   3. If \(a = b\):

      The positive and the negative cancel out and the result is zero:

      \(a + (-a) = 0\).

      Example: \(5 + (-5) = 0\).

4. Summary rule.

   In words: a positive plus a negative equals the difference of their absolute values, with the sign of the number that has the larger absolute value. Symbolically, for positive \(a\) and \(b\):

   \(a + (-b) =\)

   \[
   \begin{cases}
   a – b, & \text{if } a > b \\
   0, & \text{if } a = b \\
   -(b – a), & \text{if } a < b
   \end{cases}
   \]