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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.

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  • $\begingroup$ You can write it as $30=10+10+10$. $\endgroup$ – dxiv Nov 17 '16 at 18:34
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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.

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  • $\begingroup$ I like your answer and the comment of @dvix . From what I know about it's history, the mathematics is a unity of many different modeling systems. Because operations (+, -, *, /) were created at different times (as well as any part of the matematics) it's logical that they represent different "ideas". It's amazing how all this different modeling system have converged into a single entity. The answer to my question lies in the comment mentioned and the reason why my question seeks a wrong interpretation of an operation is in the answer above. $\endgroup$ – Yordan Nov 17 '16 at 19:23
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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 $b \cdot x = a$.

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  • $\begingroup$ Yes and no. Division is sometimes an operation on collections. That's how kids correctly describe distributing candy among friends. It's also the inverse of multiplication. Fractions can be understood both as parts of a whole and as numbers to be manipulated arithmetically. Most people routinely and unconsciously choose the model appropriate to a particular situation. But the fact that there are several useful models makes teaching fractions to kids difficult. See math.stackexchange.com/questions/1127483/… $\endgroup$ – Ethan Bolker Dec 9 '17 at 14:18

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