This fundamental proof is really bothering me for a long time. I have seen the proofs on proofwiki and other sites but it uses too much mathematical jargon. I would like a nice, intuitive proof using only basic arithmetic axioms and induction.
P.S (If it is not possible to prove this fact for real numbers at least give a proof for natural numbers.
Thank You!
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Shaunak Apte
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See this post – Mauro ALLEGRANZA Oct 27 '20 at 08:05
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And see this post for commutativity of multiplication – Mauro ALLEGRANZA Oct 27 '20 at 08:06
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But Peano axioms are for naturals and Induction also works for naturals: it relies on the immediate successor of a number $bn$ and in the real domain there is no immediate successor of $r$. – Mauro ALLEGRANZA Oct 27 '20 at 08:09
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An see this post for commutativity of addition – Mauro ALLEGRANZA Oct 27 '20 at 08:19
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Commutativity for multiplication is quite simple; consider a rectangle of base A and side B. Its area is $\text {length}(A) \times \text {length}(B)$. Now capsize it: the base is now B and A is the new side. What happened to the area ? – Mauro ALLEGRANZA Oct 27 '20 at 10:35
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Commutativity for addition is also very simple: put 5 dollars in your pocket and then put 4 dollars. Then empty the pocket and start again: first 4 dollars and then 5 dollars. – Mauro ALLEGRANZA Oct 27 '20 at 10:37