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For natural numbers
If a and d are natural numbers, with d non-zero, it can be proven that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < d. The number q is called the quotient, while r is called the remainder. See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder.
General case for integers
If a and d are integers, with d non-zero, then a remainder is an integer r such that a = qd + r for some integer q, and with |r| < |d|.
When defined this way, there are two possible remainders. For example, the division of −42 by −5 can be expressed as either
- −42 = 9×(−5) + 3
as is usual for mathematicians, or
- −42 = 8×(−5) + (−2).
So the remainder is then either 3 or −2.
This ambiguity in the value of the remainder can be quite serious computationally; for mission critical computing systems, the wrong choice can lead to dangerous consequences. In the case above, the negative remainder is obtained from the positive one just by subtracting 5, which is d. This holds in general. When dividing by d, if the positive remainder is r1, and the negative one is r2, then
- r1 = r2 + d.
For floating-point numbers
When a and d are floating-point numbers, with d non-zero, a can be divided by d without remainder, with the quotient being another floating-point 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 floating-point remainder r such that a = qd + r with 0 ≤ r < |d|. As in the case of division of integers, the remainder could be required to be negative, that is, −|d| < r ≤ 0.
Extending the definition of remainder for floating-point numbers as described above is not of theoretical importance in mathematics; however, many programming languages implement this definition—see modulo operation.
Satisfaction of an inequality
The way remainder was defined, in addition to the equality a = qd + r an inequality was also imposed, which was either 0 ≤ r < |d| or −|d| < r ≤ 0. Such an inequality is necessary in order for the remainder to be unique—that is, for it to be well-defined. The choice of such an inequality is somewhat arbitrary. Any condition of the form x < r ≤ x + |d| (or x ≤ r < x + |d|), where x is a constant, is enough to guarantee the uniqueness of the remainder.
In programming languages
With two choices for the inequality, there are two choices for the remainder, one negative and the other positive. This means that there are also two possible choices for the quotient. In number theory the positive remainder is chosen by convention. But programming languages need not, and different languages have adopted different conventions: Pascal goes with number theory in choosing the result of the mod operation positive (while not always a = (a div d )*d + a mod d ). C99 chooses the remainder with the same sign as the dividend a. (Before C99, the C language allowed either choice.) Perl, Python (only modern versions), and Common Lisp choose the remainder with the same sign as the divisor d. Haskell and Scheme offer two functions, remainder and modulo – PL/I has mod and rem, while Fortran has mod and modulo; in each case, the former agrees in sign with the dividend, and the latter with the divisor.
- Pascal ISO 7185:1990 22.214.171.124
- Davenport, Harold (1 x999). The higher arithmetic: an introduction to the theory of numbers. Cambridge, UK: Cambridge University Press. p. 25. ISBN 0-521-63446-6.
- Katz, Victor, ed. (2007). The mathematics of Egypt, Mesopotamia, China, India, and Islam : a sourcebook. Princeton: Princeton University Press. ISBN 9780691114859.
- Schwartzman, Steven (1994). "remainder (noun)". The words of mathematics : an etymological dictionary of mathematical terms used in english. Washington: Mathematical Association of America. ISBN 9780883855119.
- Smith, David Eugene (1958). History of Mathematics, Volume 2 (1st ed.). New York: Courier Dover. ISBN 9780486204307.
- Zuckerman, Martin M. Arithmetic: A Straightforward Approach. Lanham, Md: Rowman & Littlefield Publishers, Inc. ISBN 0-912675-07-1.