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In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder —an amount "left over"— is also acknowledged. In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In mathematics, a quotient is the end result of a division problem. ...
The remainder for natural numbers If a and d are natural numbers, with d non-zero, it can be proved 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. The division algorithm provides a proof of this result and also an algorithm describing how to calculate the remainder. A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
The division algorithm is a theorem in mathematics which precisely expresses the outcome of the usual process of division of integers. ...
Examples - When dividing 13 by 10, 1 is the quotient and 3 is the remainder, because 13=1×10+3.
- When dividing 26 by 4, 6 is the quotient and 2 is the remainder, because 26=6×4+2.
- When dividing 56 by 7, 8 is the quotient and 0 is the remainder, because 56=7×8+0.
The case of general 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 0 ≤ |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
or - −42 = 8×(−5) + (−2).
So the remainder is then either 3 or −2.
This ambiguity in the value of the remainder is not very serious; 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.
The remainder for real numbers When a and d are real numbers, with b non-zero, a can be divided by b 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 < |d|. As in the case of division of integers, the remainder could be required to be negative, that is, -|d| < r ≤ 0. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Extending the definition of remainder for real numbers as described above is not of theoretical importance in mathematics; however, many programming languages implement this definition — see modulo operation. Computer code (HTML with JavaScript) in a tool that uses syntax highlighting (colors) to help the developer see the purpose of each piece of code. ...
In computing, the modulo operation finds the remainder of division of one number by another. ...
The inequality satisfied by the remainder 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.
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