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In real analysis books, it is taken as axiom. But I want to know the logic behind it. As for example, 32*75 means adding 32 for 75 times and 75*32 means adding 75 for 32 times. How to be sure these two quantities are equal? Please mention any book which discusses the logic behind basic properties of addition, subtraction, multiplication, division. I want to know about history and development of arithmetic. Please mention any relevant books.

DREAM ISI-CMI-IIT
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    For rational numbers it holds. It would be nice if it would to hold for real numbers as well. If it's taken as axiom in your book then there's really not much left to say – Jakobian Apr 09 '19 at 18:01
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    Note that by saying "adding a to itself, b times" you implicitly assumes that $b$ is a natural number. – Yanko Apr 09 '19 at 18:02
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    If you have a rectangle with dimensions $32 \times 75$, its area stays the same when you rotate it by $90^\circ$. – Jair Taylor Apr 09 '19 at 18:03
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    More formally, every real number is the limit of a convergent sequence of rational numbers and multiplication of (existing) limits is commutative. – B.Swan Apr 09 '19 at 18:10
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    Actually the crucial part is commutativity for natural numbers as the rest follows as above and this is proved by induction from basic Peano axioms though first one uses induction to prove commutativity of addition which is a little tricky; it is actually a useful though somewhat tedious exercise to do when one starts learning basic naive set theory a la Halmos which is the book to learn these basics; using Dedekind cuts to define reals makes multiplication more challenging than using series and that's another cool exercise – Conrad Apr 09 '19 at 18:25

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