Remainder

In arithmetic, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder—an amount "left over" after the division—is also accepted.

 The remainder for natural numbers 
Given a natural number  and a non-zero natural number , it can be shown that there exist unique integers  and , such that  and . The number  is called the quotient, while  is called the remainder.

The case of general integers 
If  and  are integers with  being non-zero, then a remainder is an integer  such that  for some integer , and with .

When defined this way, there are two possible remainders. For example, the division of -42 by -5 can be expressed as either

or

.

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 . This holds in general. When dividing by , if the positive remainder is , and the negative one is , then

.

 The remainder for real numbers
When  and  are real numbers, with  being non-zero, then  can be divided by  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 proven that there exists a unique integer quotient  and a unique real remainder  such that  with . Similar to the case of division of integers, the remainder could be required to be negative, that is,  . 

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 modulation for more.

Related pages
Divisor, a number which evenly divides another
Long division
Modular arithmetic
Modulation

 References 

Category:Arithmetics