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I was reading this answer to the question of the division by zero which proposes the idea of "warped" numbers.

So, am I correct, the following rules could be a definition of division by zero:

Let:

  • $\infty_w$ be a type of infinity with the properties below.
  • $Q+$ be the set of rational numbers and $\infty_w$
  • $x$ be any element of $Q+$

(1) $\frac{x}{0} = x\times\frac{1}{0}$

(2) $\frac{1}{0} = \infty_w$

(3) $x + \infty_w = \infty_w$

(4) $x$ if nonzero: $x \times \infty_w = \infty_w$

(5) $0 \times \infty_w = 0$

(6) $\frac{1}{\frac{1}{0}} = 0$

The problem is that there is no gain from such a logic. Division by zero would only be interesting if there was some nontrivial theorem that could be attained from such a logic?

Did I get this right or is the above example different from "warped" numbers presented in the previous answer? Or are there more contradictions that arise from the example above?

My main point is understanding the strange logic that is required for division by zero to be logically consistent and how these assumptions would need to break basic assumptions about rational numbers. Is my example above logically valid?

Larry Freeman
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    $0 = (1 - 1)$ so what is $(1-1)\times \infty_w$? Did we lose distributivity? – JMoravitz Feb 26 '24 at 20:40
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    But $(a+b)\times c = (a\times c) + (b\times c)$ would be $\infty_w + \infty_w$ here – JMoravitz Feb 26 '24 at 20:44
  • Great point. $(1-1)\times \infty_w = \infty_w - \infty_w = x + \infty_w$ So it is necessary for $\infty_w - \infty_w = 0$ but this breaks all the assumptions that I am showing. Cool! Thanks! – Larry Freeman Feb 26 '24 at 20:44
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    As you look long enough you will find that there are many problems and either we lose several incredibly basic properties that we want to have... causing things like commutativity, distributivity, cancellation properties... even basic equality in some cases to break, or we keep bending over backwards to try to define edge cases over and over again to try to avoid this. In the end, it is not useful for solving any problems. Definitions are not typically written for definition's sake... the problems come first and definitions are written in a way to solve the problems they are meant to. – JMoravitz Feb 26 '24 at 20:45
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    That... or of course the other possibility is that everything collapses and all numbers in your system wind up being equal. – JMoravitz Feb 26 '24 at 20:46
  • So, if we introduce this, we need to throw out the distributive law. The distributive law is what causes all the contradictions but without it, all the numbers in the system wind up being equal which is worthless. :-) – Larry Freeman Feb 26 '24 at 20:51
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    You might be interested in wheel theory – J. W. Tanner Feb 26 '24 at 20:52
  • Thanks! I'll check out Wheel Theory. Looks very interesting. – Larry Freeman Feb 26 '24 at 21:24
  • Just define it as reverse multiplication. 8 $\div$ 0 is asking "what number times 0 equals 8?". – Nate Feb 27 '24 at 02:11
  • @Nate check out wheel theory. Reverse multiplication to give 8 leads inevitably to contradictions because of the distributive law. Wheel theory has a definition which proposes a special value to cover division by 0 but it does not support reverse multiplication. – Larry Freeman Feb 28 '24 at 10:48

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