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

Solution 1:

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.

Solution 2:

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

Solution 3:

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.

Solution 4:

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.

:)

Solution 5:

Combining my comments into an answer:

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.)