Arithmetic with Signed Numbers

Adding and subtracting signs

Signed-number arithmetic becomes easy once you split addition into two cases. If the two numbers have the same sign, add their absolute values and keep that sign: (−6) + (−2) = −8. If they have opposite signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger size: (−6) + 2 = −4, while (−2) + 6 = 4.

Subtraction needs no new rules at all. To subtract a number, add its [[additive-inverse|additive inverse]] (its opposite). So 5 − 8 means 5 + (−8) = −3, and 4 − (−3) means 4 + (+3) = 7. The phrase “subtracting a negative is adding” is just this rule in plain words — and it is exactly the inverse operation view: subtraction undoes addition by adding the opposite.

Subtraction = add the opposite

   5  -  8           ->   5 + (-8)  = -3
   4  - (-3)         ->   4 + (+3)  =  7
  -7  -  2           ->  -7 + (-2)  = -9
  -7  - (-10)        ->  -7 + (+10) =  3

Multiplying and dividing signs

For multiplication and division, the size is just the product or quotient of the absolute values; only the sign needs care. Like signs give a positive; unlike signs give a negative. So (−4)(−5) = 20, (−4)(5) = −20, and −20 ÷ 5 = −4. Why does a negative times a negative turn positive? Watch the pattern as one factor drops: 3·(−2) = −6, 2·(−2) = −4, 1·(−2) = −2, 0·(−2) = 0, so (−1)·(−2) must continue the climb to +2.

1. Compute the answer’s size by multiplying (or dividing) the absolute values: |−4|·|−5| = 20.
2. Count the negative factors. An even count gives a positive result; an odd count gives a negative one.
3. Attach the sign: (−4)(−5) has two negatives → positive 20.