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It is possible (and common when explaining multiplication) to express multiplication of positive integers as repeated addition like this:

$$4\cdot5 = 5+5+5+5=20$$

And a general expression would be this:

$$a\cdot b=\sum_{n=1}^ab$$

Is the reverse possible? Is it possible to express addition of positive integers as repeated multiplication?

$$a+b=\prod_{n=?}^??$$

Edit: This question is not a duplicate of How to define addition through multiplication? because my question does not put as strong limitations on what repeated multiplication means. A trivial answer would be $1\cdot(a+b)$, but is not so interesting. I found @barakmanos answer interesting enough.

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    $\begingroup$ see math.stackexchange.com/questions/350469/… $\endgroup$ – User Jan 26 '15 at 20:51
  • $\begingroup$ What are we allowed to use inside the $\prod$? Is a minus sign ok? $\endgroup$ – barak manos Jan 26 '15 at 20:58
  • $\begingroup$ Let's say any real factor. $\endgroup$ – Jostein Trondal Jan 26 '15 at 21:01
  • $\begingroup$ I was referring to arithmetic operations. For example, can we use $\prod\limits_{?}^{?}a-b$? $\endgroup$ – barak manos Jan 26 '15 at 21:03
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    $\begingroup$ Choose any $n\in\mathbb{N}$ you like, then $a+b=\prod\limits_{i=1}^{n}\sqrt[n]{\frac{a^2-b^2}{a-b}}$ $\endgroup$ – barak manos Jan 26 '15 at 21:09
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Choose any $n\in\mathbb{N}$ you like: $$a+b=\prod_{i=1}^{n}\sqrt[n]{\frac{a^2-b^2}{a-b}}$$

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  • $\begingroup$ This is recursive. $a - b$ is an addition. $\endgroup$ – rubik Jan 26 '15 at 21:48
  • $\begingroup$ @rubik: So is ${a}\cdot{b}$, which also relies on addition. In fact, it is pretty obvious even in the example that OP gave within the question itself, of ${a}\cdot{b}=\sum\limits_{n=1}^{a}b$. Hence $a^b$ is also based on addition, and of course, $\ln({e^a}\cdot{e^b})$ (given in the link suggested in the first comment to the question), which is calculated as an infinite series consisting of several such arithmetic operations. You can say that pretty much every arithmetic operation is eventually based on a finite or infinite number addition operations. Please revoke your down-vote. $\endgroup$ – barak manos Jan 26 '15 at 22:02
  • $\begingroup$ If you see the link that Anders Mariegaard posted it is clear that what OP wants is impossible, because every answer would be recursive. So a correct answer would state that no, it is not possible to define addition as repeated multiplication because such a definition would be self-referencing and thus not a valid definition. $\endgroup$ – rubik Jan 26 '15 at 22:09
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    $\begingroup$ Dear @rubik. I can assure you that the question is not misphrased. I carefully phrased it. I did not put strong limitations on what I meant when I said repeated multiplication. The accepted answer uses repeated multiplication. You cannot contest that. But I know what you mean, and what you mean is a stronger limitation on what repeated multiplication means. The OP in Anders' link says "But suppose we are in a universe where we can only multiply". I did not want to put that strong limitations in my question. Hence, my question is not a duplicate. And please revoke your down-vote. $\endgroup$ – Jostein Trondal Jan 27 '15 at 9:28
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    $\begingroup$ Since the OP is content with the answer and I am no math professor I edited the answer so that I could remove the down-vote. I still believe that this is not a valid answer though. $\endgroup$ – rubik Jan 27 '15 at 14:01

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