When multiplication is defined on objects other than the real numbers, is there an attempt to define this operation in terms of addition?

Question

I do not understand "abstract" multiplication. If for integers it is repeated addition (in n*k, n times where n is an integer), then what could it even mean for something as simple as multiplying two irrational numbers? Or for more exotic things like vectors? Why for example, is vector addition called addition rather than vector multiplication and vice versa? Perhaps this is not the best expressed question but I hope people know what I am asking.

Answer 1

This is, in a way, more about the language of math than about math itself. But language also carries a lot of mathematical meaning. In general, we have many different algebraic structures with two binary operations defined on them. One of those is usually commutative and serves as the "base" operation. We call that one a sum, or addition, and write it as $+$ or some variation thereof like $\oplus$. Then there is another operation which plays nicely with the base operation $+$. Playing nicely means that it distributes over $+$. We call that one a product, or multiplication, and write it in a variety of different ways ($\cdot,\times,(\cdot,\cdot),\langle\cdot,\cdot\rangle$, etc.). The fact that it distributes over addition is a direct generalization of the "repeated addition" thing in the integers. The fact that multiplication distributes over addition means that $n\cdot m=(1+\dots+1)\cdot m=m+\dots+m$. Distributivity is a way to enforce this "repeated addition" concept by saying that if $n$ can be obtained by repeatedly adding $1$, then $nm$ can be obtained by repeatedly adding $m$. But the distributive property holds for numbers which can't be obtained by repeatedly adding $1$, so it's a generalization. Though we could still say that if $a$ can be obtained by repeatedly adding $b$ (so $a=b+\dots+b$), then $ax$ can be obtained by repeatedly adding $bx$. Of course, we have to define what $bx$ is, but in a way, the distributive property still enforces a certain concept of repeated addition.

Anyway, now that distributivity over addition has been identified as a core feature of products, we tend to call everything that distributes over a commutative, associative binary operation ($+$) a product or multiplication. The product of a field distributes over the addition of the field, the scalar multiplication of a vector space distributes over its addition, the inner product and the vector products do, too. Matrix products distribute over matrix sums. And so on.