Fractions
A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: 1/2 and 17/3) consists of an integer numerator, displayed above a line (or before a slash like 1⁄2), and a non-zero integer denominator, displayed below (or after) that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.
Fractions can be used to represent ratios and division. Thus the fraction 3/4 can be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four).
We can also write negative fractions, which represent the opposite of a positive fraction. For example, if 1/2 represents a half-dollar profit, then −1/2 represents a half-dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −1/2, −1/2 and 1/−2 all represent the same fraction – negative one-half. And because a negative divided by a negative produces a positive, −1/−2 represents positive one-half.
In mathematics a rational number is a number that can be represented by a fraction of the form a/b, where a and b are integers and b is not zero; the set of all rational numbers is commonly represented by the symbol
Q
{\displaystyle \mathbb {Q} }
or Q, which stands for quotient. The term fraction and the notation a/b can also be used for mathematical expressions that do not represent a rational number (for example
2
2
{\displaystyle \textstyle {\frac {\sqrt {2}}{2}}}
), or even do not represent any number (for example the rational fraction
1
x
{\displaystyle \textstyle {\frac {1}{x}}}
).
A rational number, expressed as
p
q
{\displaystyle {\frac {p}{q}}}
where p and q are coprime integers and is in base
b
{\displaystyle {b}}
, has a terminating representation in base
b
{\displaystyle {b}}
if and only if q divides a power of b, or
p
q
=
C
b
n
{\displaystyle {\frac {p}{q}}={\frac {C}{b^{n}}}}
,for some
C
{\displaystyle {C}}
and some integer
n
{\displaystyle {n}}
> 0.
By cross multiplying, the equality is equivalent to
q
C
=
p
b
n
{\displaystyle {qC}={pb^{n}}}
. Because q doesn't divide p , it must divide
b
n
{\displaystyle b^{n}}
, and the expansion will not continue.
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