# Fraction

[Lat.=breaking], in arithmetic, an expression representing a part, or several equal parts, of a unit.

## Notation for Fractions

In writing a fraction, e.g., 2/5 or 2/5, the number after or below the bar represents the total number of
parts into which the unit has been divided. This number is called the denominator. The number before or
above the bar, the numerator, denotes how many of the equal parts of the unit have been taken. The expression 2/5,
then, represents the fact that two of the five parts of the unit or quantity have been taken. The present notation
for fractions is of Hindu origin, but some types of fractions were used by the Egyptians before 1600 B.C. Another
way of representing fractions is by decimal notation (see decimal system).

## Characteristics of Fractions

When the numerator is less than the denominator, the fraction is proper, i.e., less than unity. When the reverse is
true, e.g., 5/2, the fraction is improper, i.e., greater than unity. When a fraction is written with a whole number,
e.g., 31/2, the expression is called a mixed number. This may also be written as an improper fraction, as 7/2, since
three is equal to six halves, and by adding the one half, the total becomes seven halves, or 7/2. A fraction has been
reduced to its lowest terms when the numerator and denominator are not divisible by any common divisor except 1, e.g.,
when 4/6 is reduced to 2/3.

## Arithmetic Operations Involving Fractions

When fractions having the same denominator, as 3/10 and 4/10, are added, only the numerators are added, and their
sum is then written over the common denominator: 3/10+4/10=7/10. Fractions having unlike denominators, e.g., 1/4
and 1/6, must first be converted into fractions having a common denominator, a denominator into which each denominator
may be divided, before addition may be performed. In the case of 1/4 and 1/6, for example, the lowest number into which
both 4 and 6 are divisible is 12. When both fractions are converted into fractions having this number as a denominator,
then 1/4 becomes 3/12, and 1/6 becomes 2/12. The change is accomplished in the same way in both cases: the denominator
is divided into the 12 and the numerator is multiplied by the result of this division. The addition then is performed as
in the case of fractions having the same denominator: (1/4)+(1/6)=(3/12)+(2/12)=5/12. In subtraction, the numerator and the denominator
are subjected to the same preliminary procedure, but then the numerators of the converted fractions are subtracted: (1/4)-(1/6)=(3/12)-(2/12)=1/12.

In multiplication the numerators of the fractions are multiplied together as are the denominators without needing change:
(2/3)×(3/5)=6/15. It should be noted that the result, here 6/15, may be reduced to 2/5 by dividing both numerator and
denominator by 3. The division of one fraction by another, e.g., (3/5):(1/2), is performed by inverting the divisor
and multiplying: (3/5):(1/2)=(3/5)×(2/1)=6/5. The same rules apply to the addition, subtraction, multiplication, and
division of fractions in which the numerators and denominators are algebraic expressions.

source: The Columbia Encyclopedia, Sixth Edition. 2001.