Editing Division (mathematics) (section)

=== Of integers ===
Integers are not [[Closure (mathematics)|closed]] under division. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches:
# Say that 26 cannot be divided by 11; division becomes a [[partial function]].
# Give an approximate answer as a [[floating-point number]]. This is the approach usually taken in [[numerical computation]].
# Give the answer as a [[fraction (mathematics)|fraction]] representing a [[rational number]], so the result of the division of 26 by 11 is <math>\tfrac{26}{11}</math> (or as a [[mixed number]], so <math>\tfrac{26}{11} = 2 \tfrac 4{11}.</math>) Usually the resulting fraction should be simplified: the result of the division of 52 by 22 is also <math>\tfrac{26}{11}</math>. This simplification may be done by factoring out the [[greatest common divisor]].
# Give the answer as an integer ''[[quotient]]'' and a ''[[remainder]]'', so <math>\tfrac{26}{11} = 2 \mbox{ remainder } 4.</math> To make the distinction with the previous case, this division, with two integers as result, is sometimes called ''[[Euclidean division]]'', because it is the basis of the [[Euclidean algorithm]].
# Give the integer quotient as the answer, so <math>\tfrac{26}{11} = 2.</math> This is the ''[[floor function]]'' applied to case 2 or 3. It is sometimes called '''integer division''', and denoted by "//".

Dividing integers in a [[computer program]] requires special care. Some [[programming language]]s treat integer division as in case 5 above, so the answer is an integer. Other languages, such as [[MATLAB]] and every [[computer algebra system]] return a rational number as the answer, as in case 3 above. These languages also provide functions to get the results of the other cases, either directly or from the result of case 3.

Names and symbols used for integer division include {{mono|div}}, {{mono|/}}, {{mono|\}}, and {{mono|%}}.{{cn|reason=Give a reference for each of the (claimed) names. I myself never sav '\' or '%' denoting division.|date=February 2025}} Definitions vary regarding integer division when the dividend or the divisor is negative: [[rounding]] may be toward zero (so called T-division) or toward [[Extended real number line|−∞]] (F-division); rarer styles can occur – see [[modulo operation]] for the details.

[[Divisibility rule]]s can sometimes be used to quickly determine whether one integer divides exactly into another.