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Comment by aaplok | Hacker News Reader

Multiplication is literally repeated addition, as in "taking a number multiple times". It is in the name.

The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation is expected to leave that property intact.

This is not unique to multiplication. Modern mathematics is all about extracting fundamental properties that make something true, to generalize results while keeping the original simpler statement true. Starting from the generalisations without explaining the intellectual process it took us to get there means just giving facts without explaining them.

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"Multiplication is literally repeated addition"

How do you repeat something a negative number of times? What does it mean to multiply two negatives, if multiplication is repeated addition? This is not an overly generalized concept, we teach negative numbers to elementary school students.

I don't know what you mean by "overly". Multiplication by negative numbers is a generalisation of the earlier concept of multiplication by positive numbers. In a sense even multiplying by 1 is a generalisation (at least if you read Euclid he doesn't really define multiplying by 1).

I don't claim that we should not teach generalisations to children. What I claim is that of we teach generalisations we at least should be aware that they are generalisations, and maybe even tell them about it.

That sounds a bit backwards to me. If anything, negative numbers are the generalization (as is 0), and we should have a concept of multiplication that works for such generalizations. Euclid's approach to arithmetic always seemed strained to me, something motivated by a religious view of math where a compass and straightedge were the only tools anyone should need.

Repeated addition is just a technique for multiplication. It is one of many techniques we teach children, and we should leave it at that -- a technique. When we get to multiplication by negative numbers or fractions, we teach children other techniques, and we do not feel any need to try to "define" multiplication in terms of those techniques. Why should we "define" multiplication in terms of repeated addition, and then do circles around ourselves trying to generalize that "definition?" We can just say that multiplication is one operation we can do on numbers; addition is another, and they are related by the distributive law (and it from the distributive law that we can derive the various techniques we use for multiplication).

Define π times e in terms of repeated addition.

I wouldn't say multiplication is literally repeated addition. I'd say it reduces to repeated addition when the multiplier is a natural number.

Maybe it is not literally addition but it certainly reduces to addition in all cases: https://news.ycombinator.com/item?id=26329855

Nope. I'll reply there.

I am afraid I did not make myself clear. I don't mean that multiplying by irrational numbers is repeated addition. What I mean is that multiplication by irrational (or even non-integers) was defined to generalize the earlier notion of multiplying by positive integers.

Also I mean "literally" literally :). Etymologically, Multiplication is the act of taking/creating many (multi in Latin) copies of something (as in "multiplying breads").

That these generalizations don't fit the concept of repeated addition doesn't erase the fact that the primitive concept meant precisely that. Whether we decide to hide that fact from children can be a conscious decision, but at least it should be deliberate.

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