## 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: ${\displaystyle {\tfrac {1}{2}}}$ and 17/3) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. The picture to the right illustrates ${\displaystyle {\tfrac {3}{4}}}$ or ¾ of a cake.

Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and 10−2 respectively, all of which are equivalent to 1/100). An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1.

${\displaystyle a\times {\frac {b}{c}},}$
${\displaystyle a\cdot {\frac {b}{c}},}$
${\displaystyle a\left({\frac {b}{c}}\right).}$
Not to be confused with fractions involving complex numbers
${\displaystyle {\frac {\tfrac {1}{2}}{\tfrac {1}{3}}}={\tfrac {1}{2}}\times {\tfrac {3}{1}}={\tfrac {3}{2}}=1{\tfrac {1}{2}}}$
${\displaystyle {\frac {12{\tfrac {3}{4}}}{26}}=12{\tfrac {3}{4}}\cdot {\tfrac {1}{26}}={\tfrac {12\cdot 4+3}{4}}\cdot {\tfrac {1}{26}}={\tfrac {51}{4}}\cdot {\tfrac {1}{26}}={\tfrac {51}{104}}}$
${\displaystyle {\frac {\tfrac {3}{2}}{5}}={\tfrac {3}{2}}\times {\tfrac {1}{5}}={\tfrac {3}{10}}}$
${\displaystyle {\frac {8}{\tfrac {1}{3}}}=8\times {\tfrac {3}{1}}=24.}$
${\displaystyle {\frac {5}{10/{\tfrac {20}{40}}}}={\frac {1}{4}}\quad }$ or ${\displaystyle \quad {\frac {\tfrac {5}{10}}{\tfrac {20}{40}}}=1}$
${\displaystyle {\tfrac {63}{462}}={\tfrac {63\div 21}{462\div 21}}={\tfrac {3}{22}}}$
${\displaystyle {\tfrac {3}{4}}>{\tfrac {2}{4}}}$ because 3>2.
${\displaystyle {\tfrac {2}{3}}}$ ? ${\displaystyle {\tfrac {1}{2}}}$ gives ${\displaystyle {\tfrac {4}{6}}>{\tfrac {3}{6}}}$
${\displaystyle {\tfrac {5}{18}}}$ ? ${\displaystyle {\tfrac {4}{17}}}$
${\displaystyle {\tfrac {5\times 17}{18\times 17}}}$ ? ${\displaystyle {\tfrac {4\times 18}{17\times 18}}}$
${\displaystyle {\tfrac {2}{4}}+{\tfrac {3}{4}}={\tfrac {5}{4}}=1{\tfrac {1}{4}}}$.
${\displaystyle {\frac {1}{4}}\ +{\frac {1}{3}}={\frac {1\times 3}{4\times 3}}\ +{\frac {1\times 4}{3\times 4}}={\frac {3}{12}}\ +{\frac {4}{12}}={\frac {7}{12}}.}$
${\displaystyle {\frac {3}{5}}+{\frac {2}{3}}}$
${\displaystyle {\frac {3}{5}}+{\frac {2}{3}}}$
${\displaystyle {\frac {9}{15}}+{\frac {10}{15}}={\frac {19}{15}}=1{\frac {4}{15}}}$
${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+cb}{bd}}}$
${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}+{\frac {e}{f}}={\frac {a(df)+c(bf)+e(bd)}{bdf}}}$
${\displaystyle {\frac {3}{4}}+{\frac {5}{12}}={\frac {9}{12}}+{\frac {5}{12}}={\frac {14}{12}}={\frac {7}{6}}=1{\frac {1}{6}}}$
${\displaystyle {\tfrac {2}{3}}-{\tfrac {1}{2}}={\tfrac {4}{6}}-{\tfrac {3}{6}}={\tfrac {1}{6}}}$
${\displaystyle {\tfrac {2}{3}}\times {\tfrac {3}{4}}={\tfrac {6}{12}}}$
${\displaystyle {\tfrac {2}{3}}\times {\tfrac {3}{4}}={\tfrac {{\cancel {2}}^{~1}}{{\cancel {3}}^{~1}}}\times {\tfrac {{\cancel {3}}^{~1}}{{\cancel {4}}^{~2}}}={\tfrac {1}{1}}\times {\tfrac {1}{2}}={\tfrac {1}{2}}}$
${\displaystyle 6\times {\tfrac {3}{4}}={\tfrac {6}{1}}\times {\tfrac {3}{4}}={\tfrac {18}{4}}}$ This method works because the fraction 6/1 means six equal parts, each one of which is a whole.
${\displaystyle 3\times 2{\tfrac {3}{4}}=3\times \left({\tfrac {8}{4}}+{\tfrac {3}{4}}\right)=3\times {\tfrac {11}{4}}={\tfrac {33}{4}}=8{\tfrac {1}{4}}}$
0.5 = 5/9
0.62 = 62/99
0.264 = 264/999
0.6291 = 6291/9999
0.05 = 5/90
0.000392 = 392/999000
0.0012 = 12/9900
0.1523 + 0.0000987
1523/10000 + 987/9990000 = 1522464/9990000
x = 0.1523987
10,000x = 1,523.987
10,000,000x = 1,523,987.987
10,000,000x − 10,000x = 1,523,987.987 − 1,523.987
9,990,000x = 1,523,987 − 1,523
9,990,000x = 1,522,464
x = 1522464/9990000
${\displaystyle (a,b)+(c,d)=(ad+bc,bd)\,}$
${\displaystyle (a,b)-(c,d)=(ad-bc,bd)\,}$
${\displaystyle (a,b)\cdot (c,d)=(ac,bd)}$
${\displaystyle (a,b)\div (c,d)=(ad,bc){\text{ (when c ≠ 0)}}}$
${\displaystyle {\frac {3}{\sqrt {7}}}={\frac {3}{\sqrt {7}}}\cdot {\frac {\sqrt {7}}{\sqrt {7}}}={\frac {3{\sqrt {7}}}{7}}}$
${\displaystyle {\frac {3}{3-2{\sqrt {5}}}}={\frac {3}{3-2{\sqrt {5}}}}\cdot {\frac {3+2{\sqrt {5}}}{3+2{\sqrt {5}}}}={\frac {3(3+2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3+2{\sqrt {5}})}{9-20}}=-{\frac {9+6{\sqrt {5}}}{11}}}$
${\displaystyle {\frac {3}{3+2{\sqrt {5}}}}={\frac {3}{3+2{\sqrt {5}}}}\cdot {\frac {3-2{\sqrt {5}}}{3-2{\sqrt {5}}}}={\frac {3(3-2{\sqrt {5}})}{{3}^{2}-(2{\sqrt {5}})^{2}}}={\frac {3(3-2{\sqrt {5}})}{9-20}}=-{\frac {9-6{\sqrt {5}}}{11}}}$
६        १        २
१        १        १
४        ५        ९
6        1        2
1        1        −1
4        5        9
• 2 are white,
• 6 are red, and
• 4 are yellow,
• A unit fraction is a vulgar fraction with a numerator of 1, e.g. ${\displaystyle {\tfrac {1}{7}}}$. Unit fractions can also be expressed using negative exponents, as in 2−1, which represents 1/2, and 2−2, which represents 1/(22) or 1/4.
• An Egyptian fraction is the sum of distinct positive unit fractions, for example ${\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}}$. This definition derives from the fact that the ancient Egyptians expressed all fractions except ${\displaystyle {\tfrac {1}{2}}}$, ${\displaystyle {\tfrac {2}{3}}}$ and ${\displaystyle {\tfrac {3}{4}}}$ in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, ${\displaystyle {\tfrac {5}{7}}}$ can be written as ${\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.}$ Any positive rational number can be written as a sum of unit fractions in infinitely many ways. Two ways to write ${\displaystyle {\tfrac {13}{17}}}$ are ${\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}}$ and ${\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}}$.
• A dyadic fraction is a vulgar fraction in which the denominator is a power of two, e.g. ${\displaystyle {\tfrac {1}{8}}}$.
• special fractions: fractions that are presented as a single character with a slanted bar, with roughly the same height and width as other characters in the text. Generally used for simple fractions, such as: ½, ⅓, ⅔, ¼, and ¾. Since the numerals are smaller, legibility can be an issue, especially for small-sized fonts. These are not used in modern mathematical notation, but in other contexts.
• case fractions: similar to special fractions, these are rendered as a single typographical character, but with a horizontal bar, thus making them upright. An example would be ${\displaystyle {\tfrac {1}{2}}}$, but rendered with the same height as other characters. Some sources include all rendering of fractions as case fractions if they take only one typographical space, regardless of the direction of the bar.
• shilling or solidus fractions: 1/2, so called because this notation was used for pre-decimal British currency (£sd), as in 2/6 for a half crown, meaning two shillings and six pence. While the notation "two shillings and six pence" did not represent a fraction, the forward slash is now used in fractions, especially for fractions inline with prose (rather than displayed), to avoid uneven lines. It is also used for fractions within fractions (complex fractions) or within exponents to increase legibility. Fractions written this way, also known as piece fractions, are written all on one typographical line, but take 3 or more typographical spaces.
• built-up fractions: ${\displaystyle {\frac {1}{2}}}$. This notation uses two or more lines of ordinary text, and results in a variation in spacing between lines when included within other text. While large and legible, these can be disruptive, particularly for simple fractions or within complex fractions.
Wikipedia