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3 Responses
Here’s the street-rule I use: for a − b, start at a and walk |b| steps in the same direction as a’s sign-negatives march you left, positives march you right (so -2−5 goes 5 left to -7, -2−(-5) goes 5 left to -7, 3−(-7) goes 7 right to 10, and -3−7 goes 7 left to -10). I might be oversimplifying a bit, but this “follow the starting sign” picture keeps me from tripping.
Clean rule: In a − b, start at a and move |b| units; the sign of b alone sets the direction-left if b is positive, right if b is negative. Reason: subtraction is “add the opposite,” so a − b = a + (−b); on the number line, adding a positive goes right and adding a negative goes left. Examples: −2 − 5 → start at −2, go left 5 to −7; −2 − (−5) → go right 5 to 3; 3 − (−7) → go right 7 to 10; −3 − 7 → go left 7 to −10. Only b decides direction; a is just your starting address. If you like arrows, think of the “b arrow” and then flip its direction while keeping its length. A clear visual walkthrough is here: https://www.khanacademy.org/math/arithmetic/arith-review-negative-numbers/arith-review-adding-subtracting-negatives/a/adding-and-subtracting-negative-numbers-on-the-number-line. Does this “move |b| opposite b’s sign” picture click for you, or do you prefer always rewriting as a + (−b) first?
I trip on this too! The clean number-line rule that finally stuck for me is: in a − b, start at a, and let b alone decide your move-go the opposite way from b’s sign, by |b| steps. So if b is positive, walk left that many; if b is negative, walk right that many. a’s sign doesn’t affect the direction at all; it just sets your starting spot. This is the same as “add the opposite”: a − b = a + (−b). Examples: −2 − 5 means start at −2 and go left 5 to −7; −2 − (−5) means start at −2 and go right 5 to 3; 3 − (−7) means start at 3 and go right 7 to 10; −3 − 7 means start at −3 and go left 7 to −10. When I tell myself “let b pick the direction; ignore a for that,” my brain finally stops wobbling.