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We know that if you have 3 apples and somebody gives you 4 apples, you then have 7 apples but then if we deal with negative numbers and we have -3 apples and somebody gives us -4 apples, things can get confusing and nonsensical. How can we make sense out of integers like these?

Matthew Conroy
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Jesus Christ
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  • WW Sawyer used the example of temperature, which has a natural "below zero" on the most familiar scales - but that doesn't help with multiplication. – Mark Bennet Jun 11 '15 at 11:56
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    If you think of positive counts as "have" and negative counts as "owe" it may make more sense in some circumstances – MPW Jun 11 '15 at 12:01
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    Having -3 apples can be interpreted as owing 3 apples to someone. If you then have another -4 apples , e.g. owing another 4 apples to someone you end up with -7 apples, owing 7 apples. – Michael Stocker Jun 11 '15 at 12:03
  • Negative numbers are really just convenient for modeling certain things. You can't physically hold $-3$ apples in your hand, but you could on paper, if say you owe somebody else $3$ apples, then you would currently have a net of $-3$ apples. The negative in this case is just agreed (by all parties) to represent a debt. In that case there's no contradiction or logical fallacy. As long as we all agree on how to interpret a negative quantity then it's fine. – Gregory Grant Jun 11 '15 at 12:41

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For the example you gave, having -3 apples could be seen as owing someone 3 apples. Any apples that you get when you have a negative "apple balance" or an "apple debt" must go to the person to whom the apples are owed (until you owe no more apples, ie. you have $\geq$ 0 apples).

Gaining an apple when you have a negative apple balance reduces the amount of apples you owe by one (decreases your apple debt). If someone "gives you" a negative amount of apples when you already have a negative apple balance, your apple debt increases by the absolute value of the number of apples you were given. eg. -3 apples + -4 apples = -7 apples

You start with -3 apples, which means you have an apple debt of 3. If you gain 4 apples, you must give 3 of them away because you owe 3 apples. 4 + -3 = 1, so your apple balance is no longer negative.

Simplex
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