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3 Responses
I totally get why this feels like a double reverse, so I like to go step by step: subtraction means “add the opposite.” That rule alone separates the two cases. For -3 − (−5), I rewrite it as -3 + 5, which is 2; removing a debt of 5 pushes you 5 steps to the right from -3 to 2 on the number line. But for -3 − 5, I rewrite it as -3 + (−5), which is -8; subtracting a positive is the same as adding a negative, so you go 5 more steps left. In the cookie-debt picture: owing 3 and having 5 of that debt erased makes you end up 2 ahead, while owing 3 and then being charged 5 more deepens the debt to 8. People often say “minus minus is plus,” and I sometimes think of it as related to “two negatives make a positive,” like in multiplication, which is kind of why the signs look like they cancel here-but that mixing of multiplication and subtraction rules can be misleading if I’m not careful. Simple example: start at 4, then 4 − (−2) = 4 + 2 = 6 (two steps right), but 4 − 2 = 2 (two steps left). A nice visual explanation with number lines is here: https://www.khanacademy.org/math/arithmetic/arith-review-negative-numbers/arith-review-add-sub-negatives/v/negative-number-subtraction-intuition
I totally get this-our brains love catchy rules like “minus minus is plus,” but that phrase only applies when you’re subtracting a negative number, not every time you see two minus signs floating around! The clean rule is: subtraction means “add the opposite,” so a − b = a + (−b). That makes −3 − (−5) become −3 + 5 = 2, which matches your cookie-debt story: you owed 3, then someone cancels a debt of 5, so you end up 2 in the positive. But −3 − 5 becomes −3 + (−5) = −8: starting 3 in debt and then losing 5 more cookies means you now owe 8. Tiny analogy: imagine a number line walk-“subtract” tells you to step left, while the sign of the second number tells you which way your steps actually point; subtracting a negative flips those steps to the right, but subtracting a positive keeps you stepping left. So you’re mixing up the operation (subtract vs add) with the sign of the second number (positive vs negative)-two separate switches! For a clear refresher with visuals, this Khan Academy page is great: https://www.khanacademy.org/math/arithmetic/arith-review-negative-numbers/arith-review-add-sub-negatives/a/adding-and-subtracting-negative-numbers.
You’re mixing up the subtraction sign with the sign of the number: subtraction means “add the opposite,” so -3 − (−5) = -3 + 5 = 2 (you remove debt), but -3 − 5 = -3 + (−5) = −8 (you add more debt)-the “minus minus is plus” flip only happens when the thing you’re subtracting is itself negative. Hope this helps!