Adding and Subtracting Negative Numbers

Negative numbers count things below zero: debt, basement floors, freezing temperatures. Adding a positive moves right on the number line, adding a negative moves left. For -7 + 3, start at -7 and step 3 places right to land on -4. The sign of the answer follows whichever side has the bigger size. Subtracting flips direction, and subtracting a negative flips twice, which is why 5 - (-3) equals 5 + 3 = 8. Owing a debt and having it cancelled feels like gaining money. Try every sum as a walk: face right for plus, face left for minus, and step it out.

Same signs combine: keep the sign and add the sizes, so -2 + (-5) = -7. Different signs compete: keep the sign of the bigger size and subtract the sizes, so -7 + 3 = -4. Number lines and counters both show the same story. Counters pair positives with negatives until one side runs out. Real contexts make the rules stick. Gains and losses of game points, deposits and withdrawals, climbs and descents all add exactly like integers. Before computing, estimate: is the answer positive, negative, or zero? That one-second check catches most sign slips.

Every subtraction is addition in disguise: a - b means a + (opposite of b). So 4 - 9 becomes 4 + (-9) = -5, and 5 - (-3) becomes 5 + 3 = 8. The keep-change-change rhyme says it all: keep the first number, change minus to plus, change the second number to its opposite. Then follow the addition rules. Distance between two numbers is a subtraction with absolute value wrapped around it. From -4 to 6 the gap is |6 - (-4)| = 10. Order mixed lists by comparing signs first: any negative sits below any positive, and among negatives the bigger size is the smaller number.