Why when multiplying two negative numbers do you get a positive number?

Answer 1

The answer has to do with the fundamental properties of operations on numbers (the notions of "addition", "subtraction", "multiplication", and "division"). Each number has an "additive inverse" associated to it (a sort of "opposite" number), which when added to the original number gives zero. This is in fact the reason why the negative numbers were introduced: so that each positive number would have an additive inverse. For example, the inverse of 3 is -3, and the inverse of -3 is 3. Note that when you take the inverse of an inverse you get the same number back again: "-(-3)" means "the inverse of -3", which is 3 (because 3 is the number which, when added to -3, gives zero). To put it another way, if you change sign twice, you get back to the original sign. Now, any time you change the sign of one of the factors in a product, you change the sign of the product: (-something) × (something else) is the inverse of (something) × (something else), because when you add them (and use the fact that multiplication needs to distribute over addition), you get zero. For example, (-3) ´ (-4) is the inverse of (3) ´ (-4) because when you add them and use the distributive law, you get . (-3) ´ (-4) + (3) ´ (-4) = (-3 + 3) ´ (-4) = 0 ´ (-4) = 0 So (-3) ´ (-4) is the inverse of (3) ´ (-4) , which is itself (by similar reasoning) the inverse of 3 ´ 7. Therefore, (-3) ´ (-4) is the inverse of the inverse; in other words, the inverse of -12 in other words, 12. The fact that the product of two negatives is a positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again.