Why does dividing by a fraction make a number bigger?

Let's consider an example: 2/(1/2).

What does it mean to divide 2 by 1/2? What we're asking is, in essence, the inverse of a multiplication problem: 1/2 times what number gets us 2? Or, if we split 2 up into pieces 1/2 in size, how many pieces would make up the whole? Well, 1/2 + 1/2 + 1/2 + 1/2 = 2, which means we need four halves to make the whole (2). Similarly, if you divided 3 by 1/3, it would take 9 thirds to make three, so 3/(1/3) = 9.

It also helps to consider the fact that any number divided by 1 simply gives you back that number, and any number divided by a number greater than one produces a number less than the one you started with (i.e. 6/2 = 3, which is less than 6), so it would make sense if dividing by a number between 1 and 0 gave you back a number greater than the one you started with.