when "a" and "d" are real numbers, with "d" being non-zero, then "a" can be divided by "d" without remainder, with the quotient being another real number. if the quotient is constrained to being an integer, however, the concept of remainder is still necessary. it can be proved that there exists a unique integer quotient "q" and a unique real remainder "r" such that "a"="qd"+"r" with 0≤"r" &lt; |"d"|. similar to the case of division of integers, the remainder could be required to be negative, that is, -|"d"| &lt; "r" ≤ 0.