Say you have 3 apples and 2 oranges, and you want to multiply these two groups of fruits together to obtain a desired result, for instance:
A. You want 3 apples for each orange, so you have 6 apples and 2 oranges.
B. You want 2 oranges for each apple, so you have 6 oranges and 3 apples.
Now, lets say you want both A and B, 3 apples per 2 oranges, and 2 oranges per 3 apples, so you have 6 of each which = 12 'fruits'/objects/items etc.
This seems to be the only three multiplicative operations that can be applied when dealing with 2 groups or varying objects, and as a result, 2*3 seems to produce a different outcome than 3*2. So when working in the real world, with actual, physical objects, is multiplication ever really commutative?
Edit: So by what Sloan has shown, (hey, that rhymes!), it seems that the first scenario can be modeled by (3a*2o)/o = (6ao)/o = 6a. Where the second the variables are just reversed, 2o*3a/a = 6o.
I was confused with the third scenario however, where you just multiply 3 apples and 2 oranges together to produce 6 of each fruit, and this is why multiplication seems to be commutative, since 3*2 = 2*3.
However, it seems to be that 3*2/1 can produce different outcomes, where the distinction only makes sense when the numbers are associated with variables, like a and o.
So, it just depends on what 1 represents when used as the denominator, which to me is a very interesting property of mathematics.