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Why does multiplying two negative numbers make a positive? Well, first there is the "common sense" explanation: When I say "Eat!" I am encouraging you to eat (positive) But when I say "Do not eat!" I am saying the opposite (negative). Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).
Taking the inverse of any number is visualized by taking the mirror-image of the original plot. So the inverse of a positive number (a point to the right of zero) is a negative number (a point to the left of zero, at the same distance from zero). Likewise, the inverse of a negative number is a positive number.
If a number has no sign it usually means that it is a positive number. Example: 5 is really +5. Play with it! On the Number Line positive goes to the right and negative to the left. Try the sliders below and see what happens: Balloons and Weights. Let us think about numbers as balloons (positive) and weights (negative):
18 de mar. de 1997 · 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.
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11 de feb. de 2022 · Here's how the reasoning goes: (1) Zero times anything equals zero. (2) Every number has exactly one additive inverse. This means if N is a positive number, then -N is its additive inverse, so that N + (-N) = 0. Likewise, the additive inverse of -N is N. (3) We want negative numbers to obey the distributive law.
Expanding brackets, via the area model, gives a convincing student illustration that mathematics “wants” negative times negative to be positive. (And for students ready for it, the axiomatic approach clinches it.) As to what “negative times negative is positive” actually means – I don’t have a clue.
