Why 1 divided by 2 does not equal two halfs?

Question

I think that mathematics is a system of rules used by us to model the reality. If I divide one pile of gold in two equal piles I would get two equal piles. But in mathematics if I'm to represent the pile with a number, then the result will contain only one number. It's confusing to me that in mathematics I always get only one number, while in reality I can get any number of objects (for example, I can divide the pile of gold into 7 equal piles). I expect to see things like: 30 / 3 = [10,10,10] or stuff like that because that is more close to reality, but I always see 30 / 3 = 10. Can someone please comment on my problem.

Answer 1

You are quite right that if you have a pile of size $30$ and divide it into $3$ (equal) piles, then what you have is $3$ piles each of size $10$, which could be described as $(10,10,10)$. The mathematical notation $30/3$ refers to the size of a single pile.

Likewise, if you divide a pie into $2$ equal pieces, then what you have is not half a pie but two halves of a pie. The number $1/2$ refers to the size of each piece relative to the original whole.

Answer 2

Division is not an operation on collections; divison is the inverse operation of multiplication.

Suppose $b \neq 0$. Then $\frac{a}{b}$ is the unique $x$ such that $b \cdot x = a$.