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In Mathematics, the monoid of numbers is summation, why it can't be multiplication since both operations are monoidal (they both are associative and binary, and have an identity value)

devio
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    There is no such thing as the "monoid of (some set)", there is always only "the monoid of (some set), endowed with (some operation)". However, by abuse of language, one sometimes uses the former formulation when the intended operation is clear from the context or somehow given naturally from the set. As your question shows, such formulations can sometimes be ambiguous. – Hagen von Eitzen Mar 31 '18 at 19:57
  • @HagenvonEitzen: I see, any good resources to better understand monoids? (I'm seeking a simple way to understand the concept) – devio Mar 31 '18 at 20:39

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You are right, but in order to disambiguate, you can talk about the additive monoid of positive integers, that is, $(\mathbb{N}, +, 0)$ versus the multiplicative monoid of positive integers $(\mathbb{N}, \times, 1)$.

Answer to your comment. To start with, you can look at the Wikipedia entry Monoid.

J.-E. Pin
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