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Multiplication can be thought of as repeated addition. For instance, we have $3\times 2=6$ which is equivalently expressed as $3\times (1+1)=3+3=6$

Notice that in this definition of repeated addition, we have a 3 which comes first, to which another 3 is added. My question is this: which 3 is added first? Isn't there a "sequencing" implied here? Which 3 comes "first" to which another 3 is added? If not, how can one make sense of adding something to something else, if the something else did not exist prior to being added upon?

Thanks!

Kwame Brown
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1 Answers1

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The formal definition of multiplication of natural numbers in terms of addition can be given as: $$ \begin{align} x \cdot 0 &= 0\\ x \cdot (y + 1) &= (x \cdot y) + x \end{align} $$ This implies a certain sequence of evaluation. However, given the above definition and some standard facts about addition, one can prove that $x \cdot y = y \cdot x$, and that $x \cdot (y + z) = (x \cdot y) + (x\cdot z)$, so the actual sequence of evaluation does not matter. Read up on the Peano Axioms for more information.

This is an example of a fairly common situation where we have to give an apparently asymmetric definition of something that we know to be symmetric just to break a tie and then prove later that it didn't matter how we chose to break the tie.

Rob Arthan
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  • Thank you so much. This notion has been bugging me for a few months now- something seemed weird... – Kwame Brown Jul 29 '21 at 23:34
  • When we work in systems other than basic arithmetic, properties like this aren't always guaranteed - for example, if $A$ and $B$ are matrices, then in general $AB \neq BA$. So it often becomes important to find out which properties you can still rely on. – ConMan Jul 29 '21 at 23:39
  • @ConMan: I think it is quite clear that the OP is concerned with natural number arithmetic: there is no way in which matrix multiplication can be viewed as "repeated addition". I have made a small edit to my answer to address your point. – Rob Arthan Jul 29 '21 at 23:51
  • I know, I was looking at the more general case (which helps demonstrate why OP's question actually leads to some interesting consequences in further mathematics) – ConMan Jul 29 '21 at 23:57
  • x1 = x or x0 = 0. – marty cohen Jul 30 '21 at 00:44
  • @martycohen: thanks. I've fixed the typo. – Rob Arthan Jul 30 '21 at 00:45