Why does the dissimilar looking operation on two different sets of numbers have the same name of multiplication?

Question

The operation of incremented addition i.e. $2 \times 3$ is $2 + 2 + 2$ or $3+3$, is termed multiplication. The following operation on rational numbers $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ is also termed multiplication!

1. Why these dissimilar looking operations are termed as multiplication? In other words, what are the similarities between the two operations that allow for same name of multiplication?
2. What is the general characteristic of multiplication which helps ones think of multiplication in real numbers, complex numbers etc.? In other words, what makes multiplication, multiplication?

Answer 1

We start with arithmetic on the positive integers, and we extend this set by including it as part of a larger set, the positive rationals, and extend the operations + and x to the larger set, preserving as much as possible of their basic properties. We do further extensions: to all the rationals,to the reals, to the complex numbers. Look up the definitions of group, Abelian (commutative) group, ring, field, ordered field, complete-ordered field,and algebraically-closed field.

Answer 2

They don't look dissimilar.

$\frac a b $x$\frac c d = \frac {a \text{x} b}{c \text{x} d}= \frac{a + a + a + ... \text{ (b times)}}{c + c + c + ... \text{ (d times)}}$

2) There isn't any general thing that makes multiplication multiplication.

Depending on what math you are doing multiplications only requirement is that it be a binary operation. That is: that if you combine two elements of a set, the result is an element of the set. And that it is associative. That is (ab)c = a(bc).

There need not be any rational nor logic to the combining.

It's often combined with and operation called "addition" (which is equally arbitrary) with the condition that it is distributive. That is, a(b + c) = ab + ac.

That's it. There may or may not be that there is an element called 1 so that 1a = a for all a.

For Integers $\subset$ Rationals $\subset$ Reals $\subset$ Complex, multiplication is the same operation.

The Rationals, Reals, and Complex are what is called a field and a field has some additional multiplication axioms.

1) Multiplication is commutative: ab = ba.

2) There is a 1 such that 1a = a for all a.

3) For every $a \ne 0$ there is a $\frac 1 a$ such that $a \frac 1 a = 1$.