Mathematically speaking, the integers are not closed under divisison, meaning that the result of dividing two arbitrary integer like 3 and 5 may result in something that can not be represented by an integer. Each mathematical set of numbers (be it N, ℤ, ℚ, ℝ, ℂ) has certain limitations, and each of the representations of these sets in a computer has limitations as well. Understanding those limitations is a somewhat important point.
Wrt. division of two integers not being closed, you can deal with this in five different ways, mathematically speaking:
- Making division a partial function. This isn't desirable for a programming language that has to deal with arbitrary inputs. Do you get an arithmetic exception every time you try to divide numbers for which division is undefined? Not really a good approach.
- Make division a function from the integer to the real numbers. This is actually an approach you can use, but real numbers can not be represented exactly in computing hardware, so its only in approximate solution. Its certainly possible to implement programming languages this way, but in practice you are more likely interested in an integer result.
- Give the answer as a fractional result (Q). e.g. 3/5 becomes ⅗, sometimes represented as the tuple (3,5). You can do something like this in some programming languages, e.g. Haskell.
- Give the answer as quotient and remainder. Some programming languages do this, but Java does not support tuple types nor multiple return values, so this is out. This is sometimes called euclidean division.
- Give only the quotient as answer, discard the remainder. This is often called integer division, and is what Java implements. You can recover the remainder via a related operation, the modulo operator
%.
The JLS specifies how division works in §15.7.2:
Integer division rounds toward 0. That is, the quotient produced for operands n and d that are integers after binary numeric promotion (§5.6.2) is an integer value q whose magnitude is as large as possible while satisfying |d ⋅ q| ≤ |n|. Moreover, q is positive when |n| ≥ |d| and n and d have the same sign, but q is negative when |n| ≥ |d| and n and d have opposite signs.
There is one special case that does not satisfy this rule: if the dividend is the negative integer of largest possible magnitude for its type, and the divisor is -1, then integer overflow occurs and the result is equal to the dividend. Despite the overflow, no exception is thrown in this case. On the other hand, if the value of the divisor in an integer division is 0, then an ArithmeticException is thrown.
Moreover, you are not actually casting the result of the division to int. You are only casting index to an int. The cast is irrelevant and redundant, because integer divisions returns an int, as explained above.
