Indeed, multiplication can be understood as an abbreviation of a repetitive addition. But it can be understood in other ways, too.

For example, multiplication can be understood as a machine controlled by one factor to make the other factor bigger or smaller. From this point of view, it is no longer a matter of adding or subtracting units.

Let’s use 12 as “the other factor.” If we want to make it grow until it triples, we put three in our dial (the second factor):

12 • 3 = 36

To double its “size,” we put a 2 in our dial:

12 • 2 = 24

If we want the other factor just as it is, that is, no bigger nor smaller, we put a 1 in our dial:

12 • 1 = 12

If we want to shrink the other factor so that it is half of what it was, we put 1/2 in our dial:

12 • (1/2) = 6

Also: 12 • 0.5 = 6

If we want to shrink our other factor to a third of what it was, we put 1/3 in our dial:

12 • (1/3) = 4

So why do people express a formula as multiplications instead of additions? One answer could be: to scale things up or down proportionally.

For example, in the formula F = m•a, we express the relation between force, mass and acceleration. We’re talking about, say, a rocket in space being pushed by the constant force of the burning of it’s fuel.

Let’s solve for acceleration:

a = F / m

Here, it’s clear that the formula states that, in order to double the acceleration without altering the mass, you will need to double the constant force that is pushing the object.

Also, it’s also clear that, to double the acceleration without altering the force, you will need to be pushing half of the original mass.

As you can see, we’re talking about making the values bigger or smaller. We are not talking about repeated addition or repeated subtraction.