Is there any way to define arithmetical multiplication as other thing than repeated addition?

Is there any way to define arithmetical multiplication as other thing than repeated addition?
For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b \text{-times}}$ or $\underbrace{b+b+\cdots+b}_{a \text{-times}}$?

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Given two sets $A$ and $B$ of cardinality $a$ and $b$, respectively, the cardinality of the cartesian product $A\times B$ is called the product of $a$ and $b$, and is denoted by $a\cdot b$.

Update

When I wrote this answer I didn’t have infinite sets in mind. I just wanted to convey a mental picture of multiplication that does not involve repeated addition.

$a\cdot b$ is the the value of $f_a(b)$, where $f_a$ is the unique endomorphism of $\mathbb N$ (under addition) satisfying $f_a(1)=a$.

Here is an answer that requires an $x$ and $y$ axis. Let us say that we would like to multiply $A$ and $B$. Then we locate the point, $(1,A)$ and make the line determined by $(0,0)$ and $(1,A)$. Then locate the point $(B,0)$, and draw the vertical line that goes through this point. Then find the intersection of the vertical line just formed and the line formed by connecting the origin and $(1,A)$. You get a point, $(B,C)$, and the point $C$ is equal to $AB$.

What I like about this definition is that it works with the real numbers, the fact that it is Euclidean in spirit, and that it makes clear that multiplication is (de)magnification.

We can define $\mathbb{N}$ as the initial semiring. In this approach, not only do we not have to define multiplication as repeated addition, but, in fact, we do not have to define multiplication at all.

🙂

If you want a definition of the usual operation of multiplication that does not reduce it to repeated addition, you can always define it axiomatically the same way that we were taught it:

1. If $a, b$ are between 0 and 9, $a\cdot b$ is given by the multiplication table.
2. Otherwise, the operation is defined recursively by specifying the rules of digit-by-digit multiplication, carry, and positive/negative sign. [I don’t see the point of actually formulating the definitions here; I trust you get the idea.]

This wouldn’t be particularly insightful, since the rules appear completely arbitrary; but it is well-defined and constructive. It is, after all, the system we learned and internalized as children.

First define $a^2=\sum_{i=1}^a (2i-1)$.

Then define $a\cdot b={(a+b)^2-a^2-b^2\over 2}$.

(You could of course argue that this is equivalent to repeated addition, but the same would be true of any valid definition.)

The binary operation $\cdot$ is multiplicative if

1. There exists an $x$ such that, for every $y$, $x\cdot y=y$. That is, there is a (left?) identity element.
2. There exists an additive binary operation $+$ such that for every $x$, $y$, and $z$, $x\cdot(y+z)=x\cdot y+x\cdot z$. That is, $\cdot$ is (left?) distributive over $+$. This, of course, requires a definition of an additive operation.

This is related to the definition of a linear operator. This definition holds even for somewhat exotic definitions of multiplication, such as that for octonions (which is non-commutative and non-associative). It holds for ordinal numbers, cardinal numbers, vector spaces, projective spaces, the circle group, etc. These have interesting properties. Note that ordinal addition is itself non-commutative, for example, while the circle group possesses no zero element.

Come to think of it, where may one find examples of non-associative addition? Perhaps special relativity?

Define set $S_n$ of multiples of $n$ as intersection of all sets $S\subseteq\Bbb N$ such that $0\in S,\forall m\in S:m+n\in S$. Define least common multiple of $a,b$ as the least positive element of $S_n\cap S_m$. Define $n^2$ as a number $n$ less than lcm of $n,n+1$. Define $a\cdot b$ as half of the number $(a+b)^2-a^2-b^2$.

I like the definition of $ab$ as the area of a rectangle with side lengths $a$ and $b$. Then it’s clear that multiplication is commutative, for example.

Incidentally, this can sometimes be useful in communicating with non-mathematicians. For example, you can say that a pile of stones has a prime number of stones in it if there’s no way to arrange them into a rectangle apart from making a long line of stones. And Goldbach’s conjecture becomes: “You have a pile containing an even number of stones. Is it possible to split it into two smaller piles such that there’s no way of making a rectangle out of either of the smaller piles?”

$$a\times b = \begin{cases} \frac{a}{2} \times \left(2\times b\right) &\text{ if } a \text{ is even}\\ b +\frac{a-1}{2}\times\left(2\times b\right)& \text{ if } a \text{ is odd} \end{cases}$$

No. Because this is the fundamental definition of arithmetical multiplication, that, repeated addition is known as multiplication.

If you try to generate another definition of multiplication then you have to use this definition either directly or indirectly.