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# Why is a negative number multiplied by a negative number, a positive number?

Let's say you are playing a game involving black and red chips. At the end of the game, for each black chip that you have, you receive one dollar $$+1$$. For each red chip that you have, you have to pay one dollar $$-1$$. Now, these chips are packed together in bags of five, and say at some point in the game you've got several bags of black chips and several bags of red chips.

If someone gives you three bags of black chips, then you gain 15 dollars. $$3 \times 5 =15$$.

If someone takes away three of your bags of black chips, then you lose 15 dollars. $$-3 \times 5=-15$$.

If someone gives you three bags of red chips, then you lose 15 dollars. $$3 \times-5=-15$$.

If someone takes away three of your bags of red chips, then you gain 15 dollars. $$-3 \times -5=15$$.

The key idea is that negative numbers represent changes, not amounts. It doesn't make sense to say that you have $$-4$$ slices of bread. It does, however, make sense to say that you ate $$4$$ slices of bread, and therefore the change in the number of slices you have is $$-4$$.

Note by Raja Metronetizen
1 year, 5 months ago

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I don't know about the historic development of negative numbers. I would like to know what prompted Mathematicians to introduce them. But the concept of "debts" seems likely to be a cause of motivation or did benefit a lot as processing such matters, through negative numbers, became more effective. Amazing thing to note is that Mathematics, in this case, provided an effective notation rather than a new concept.

:) · 1 year, 5 months ago