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I know that defining division by zero is not possible because it violates the zero product property we define, that is, $0\times a=0$ for every $a$. I wonder whether we can somewhat circumvent and change the definition of the zero product property (or make a new definition about multiplying by zero), then safely deducing a definition for dividing by zero and then possibly building a system when this definition exists and proving useful theorems?

LearningMath
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  • $0 \times a = 0$ is not a definition: it's a consequence of the distributive law $a(b+c) = ab+ac$. – Crostul Sep 17 '14 at 14:34
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    Elaborating on Crostul's point, if you change $0\times a = 0$ but leave $a(b+c) = ab+ac$ alone, you're going to run into trouble. Consider for example $a\cdot 0 = a\cdot(-1 + 1) = (a\cdot -1) + (a \cdot 1) = -a + a = 0$. If you want to change $a\cdot 0 = 0$, you need to break one of the links in that chain, either $0 = -1 + 1$, or $a(b+c) = ab+ac$, or $a\cdot 1 = a$, or $a\cdot -1 = -a$, or $a + -a = 0$. But none of them seems to be dispensible. Choose your poison. – MJD Sep 17 '14 at 14:35
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    Are you just looking for a way to divide by $0$, or are you specifically asking about dividing by $0$ by changing the definition of multiplication by $0$? – Robin Goodfellow Sep 17 '14 at 14:38
  • @RobinGoodfellow Both of them. – LearningMath Sep 17 '14 at 14:39

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