integerdivision

Integer division is the operation of dividing two integers to produce an integer quotient, discarding any fractional part. In mathematics, the exact quotient is a real number, but many programming languages define an integer division operation that yields a whole-number result. The concept is often paired with a remainder, so that a = b q + r for integers a and b, where q is the quotient and r is the remainder.

There are two common conventions for integer division in programming languages. Truncation toward zero computes q by rounding the real quotient a/b toward zero, and r is defined as a − b q. This is the behavior in languages such as C, C++, and Java for integer operands. Floor division computes q by rounding a/b down to the nearest integer (toward negative infinity), with r defined so that a = b q + r; this approach is used by Python’s // operator. The two conventions yield the same result for positive operands but can differ for negative operands.

Examples:

- 7 div 3 = 2 with remainder 1 under both conventions.

- -7 div 3: trunc toward zero gives -2 with remainder -1; floor division gives -3 with remainder 2.

- 7 div -3: trunc toward zero gives -2 with remainder 1; floor division gives -3 with remainder -2.

Applications of integer division include loop control, indexing, hashing, and algorithms that require discrete steps or binning. When negative numbers are involved, programmers should be aware of the language’s specific rule set to avoid incorrect assumptions about quotients and remainders. Some languages provide both division and modulo operations with explicit, well-documented behavior to help ensure predictable results.