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We define multiplication as an operation with these properties :

a(b+c)=ab+ac and (a+b)c=ac+bc ,a*0=0 and a*1=a with a,b,c natural numbers and of course the two properties Zurtex mentioned ab=ba and a(bc)=(ab)c-I "forgot" to mention them because I didn't use them in what is included in this post . Also the naturals are produced from the function S(n)=n+1, meaning that for every n=natural, S(n)=natural. So starting from 0 and 1, we can "create" any natural.

Question: the definition of multiplication as repeatitive addition,

m*n=m+m+...+m (n m's are added) (n natural) is a result of the above definitions? I mean, the fact that every natural is equal to the sum of its previous natural and 1 shows that every natural n is an addition of n 1's. So

n=1+1+...1 (n 1's are added). Using this result and the generalisation of the distributive property we get that m*n=m*(1+1+...+1)=m+m+...+m (n times) (with m ending up as any quantity)

Are these correct? If so, do they have any actual meaning?

Thanks

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# Multiplication of integers

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