# Division Function

A division function is a rational-valued mathematical binary function that returns the quotient of two real numbers (the dividend/numerator and the divisor/denominator).

**AKA:**÷, Divide, Division, Division Operation, Ratio Function.**Example(s):****Counter-Example(s):****See:**Rational Number.

## References

### 2009

- (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Arithmetic#Division_.28.C3.B7_or_.2F.29
- Division is essentially the opposite of multiplication. Division finds the
*quotient*of two numbers, the*dividend*divided by the*divisor*. Any dividend divided by 0 is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than 1 (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend. Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is*a*÷*b*=*a*×^{1}⁄_{b}. When written as a product, it obeys all the properties of multiplication.

- Division is essentially the opposite of multiplication. Division finds the

### 2009

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Division_(mathematics)
- In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. Specifically, if c times b equals a, written: c \times b = a\, where b is not zero, then a divided by b equals c, written: \frac ab = c
(( In the above expression, a is called the dividend, b the divisor and c the quotient.

Conceptually, division describes two distinct but related settings. Partitioning involves taking a set of size a and forming b groups that are equal in size. The size of each group formed, c, is the quotient of a and b. Quotative division involves taking a set of size a and forming groups of size b. The number of groups of this size that can be formed, c, is the quotient of a and b[1].

- Teaching division usually leads to the concept of fractions being introduced to students. Unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called.

- In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. Specifically, if c times b equals a, written: c \times b = a\, where b is not zero, then a divided by b equals c, written: \frac ab = c
- (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=ratio
- S: (n) ratio (the relative magnitudes of two quantities (usually expressed as a quotient))
- S: (n) proportion, ratio (the relation between things (or parts of things) with respect to their comparative quantity, magnitude, or degree) "an inordinate proportion of the book is given over to quotations"; "a dry martini has a large proportion of gin"

- (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=rate
- S: (n) rate, charge per unit (amount of a charge or payment relative to some basis) "a 10-minute phone call at that rate would cost $5"
- S: (n) rate (a quantity or amount or measure considered as a proportion of another quantity or amount or measure) "the literacy rate"; "the retention rate"; "the dropout rate"
- …

- (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=proportion
- S: (n) proportion (the quotient obtained when the magnitude of a part is divided by the magnitude of the whole)
- …

- http://www.mathleague.com/help/ratio/ratio.htm
- A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12. We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.
- A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. 3/4 = 6/8 is an example of a proportion. When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion. Question marks or letters are frequently used in place of the unknown number.
- A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at the rate of 3 km/h. The fraction expressing a rate has units of distance in the numerator and units of time in the denominator.

Problems involving rates typically involve setting two ratios equal to each other and solving for an unknown quantity, that is, solving a proportion.