Fraction

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 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).

Negative fractions 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 ⁠${\displaystyle \mathbb {Q} }$⁠ 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 ${\displaystyle \textstyle {\frac {\sqrt {2}}{2}}}$), or even do not represent any number (for example the rational fraction${\displaystyle \textstyle {\frac {1}{x}}}$).

A rational number, expressed as ${\displaystyle {\frac {p}{q}}}$ where p and q are coprime integers and is in base ${\displaystyle {b}}$, has a terminating representation in base ${\displaystyle {b}}$ if and only if q divides a power of b, or ${\displaystyle {\frac {p}{q}}={\frac {C}{b^{n}}}}$,for some ${\displaystyle {C}}$ and some integer ${\displaystyle {n}}$ > 0.

By cross multiplying, the equality is equivalent to ${\displaystyle {qC}={pb^{n}}}$. Because q doesn't divide p , it must divide ${\displaystyle b^{n}}$, and the expansion will not continue.