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# Fractions

A fraction represents a part of a whole or, more generally, any number of equal parts. 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.

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.

## Forms of fractions

### Simple, common, or vulgar fractions

A simple fraction (also known as a common fraction or vulgar fraction) is a Rational Numbers written as a/b or $\tfrac{a}{b}$, where a and b are both integers. As with other fractions, the denominator (b) cannot be zero. Examples include $\tfrac{1}{2}$, $-\tfrac{8}{5}$, $\tfrac{-8}{5}$, $\tfrac{8}{-5}$, and 3/17. Simple fractions can be positive or negative, proper, or improper (see below). Compound fractions, complex fractions, mixed numerals, and decimals (see below) are not simple fractions, though, unless irrational, they can be evaluated to a simple fraction.

### Proper and improper fractions

Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise. In general, a common fraction is said to be a proper fraction if the  absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. It is said to be an improper fraction, or sometimes top-heavy fraction, if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, –3/4, and 4/9; examples of improper fractions are 9/4, –4/3, and 3/3.

### Mixed numbers

A mixed numeral (often called a mixed number, also called a mixed fraction) is the sum of a non-zero integer and a proper fraction. This sum is implied without the use of any visible operator such as “+”. For example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: $2+\frac{3}{4}=2\tfrac{3}{4}$.

This is not to be confused with the algebra rule of implied multiplication. When two algebraic expressions are written next to each other, the operation of multiplication is said to be “understood”. In algebra, $a \tfrac{b}{c}$ for example is not a mixed number. Instead, multiplication is understood where $a \tfrac{b}{c} = a \times \tfrac{b}{c}$.

To avoid confusion, the multiplication is often explicitly expressed. So $a \tfrac{b}{c}$ may be written as

$a \times \frac{b}{c},$
$a \cdot \frac{b}{c},$

or

$a \left(\frac{b}{c}\right).$

An improper fraction is another way to write a whole plus a part. A mixed number can be converted to an improper fraction as follows:

1. Write the mixed number $2\tfrac{3}{4}$ as a sum $2+\tfrac{3}{4}$.
2. Convert the whole number to an improper fraction with the same denominator as the fractional part, $2=\tfrac{8}{4}$.
3. Add the fractions. The resulting sum is the improper fraction. In the example, $2\tfrac{3}{4}=\tfrac{8}{4}+\tfrac{3}{4}=\tfrac{11}{4}$.

Similarly, an improper fraction can be converted to a mixed number as follows:

1. Divide the numerator by the denominator. In the example, $\tfrac{11}{4}$, divide 11 by 4. 11 ÷ 4 = 2 with remainder 3.
2. The quotient (without the remainder) becomes the whole number part of the mixed number. The remainder becomes the numerator of the fractional part. In the example, 2 is the whole number part and 3 is the numerator of the fractional part.
3. The new denominator is the same as the denominator of the improper fraction. In the example, they are both 4. Thus $\tfrac{11}{4} =2\tfrac{3}{4}$.

Mixed numbers can also be negative, as in $-2\tfrac{3}{4}$, which equals $-(2+\tfrac{3}{4}) = -2-\tfrac{3}{4}$

### Ratios

A ratio is a relationship between two or more numbers that can be sometimes expressed as a fraction. Typically, a number of items are grouped and compared in a ratio, specifying numerically the relationship between each group. Ratios are expressed as “group 1 to group 2 … to group n“. For example, if a car lot had 12 vehicles, of which

• 2 are white,
• 6 are red, and
• 4 are yellow,

then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1.

A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. We can convert these ratios to a fraction and say that 4/12 of the cars or ⅓ of the cars in the lot are yellow. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow.

### Reciprocals and the “invisible denominator”

The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of $\tfrac{3}{7}$, for instance, is $\tfrac{7}{3}$. The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverseof a fraction.

Any integer can be written as a fraction with the number one as denominator. For example, 17 can be written as $\tfrac{17}{1}$, where 1 is sometimes referred to as the invisible denominator. Therefore, every fraction or integer except for zero has a reciprocal. The reciprocal of 17 is $\tfrac{1}{17}$.

### Complex fractions

In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number,corresponding to division of fractions. For example, $\frac{\tfrac{1}{2}}{\tfrac{1}{3}}$ and $\frac{12\tfrac{3}{4}}{26}$are complex fractions. To reduce a complex fraction to a simple fraction, treat the longest fraction line as representing division. For example:

$\frac{\tfrac{1}{2}}{\tfrac{1}{3}}=\tfrac{1}{2}\times\tfrac{3}{1}=\tfrac{3}{2}=1\tfrac{1}{2}$
$\frac{12\tfrac{3}{4}}{26} = 12\tfrac{3}{4} \cdot \tfrac{1}{26} = \tfrac{12 \cdot 4 + 3}{4} \cdot \tfrac{1}{26} = \tfrac{51}{4} \cdot \tfrac{1}{26} = \tfrac{51}{104}$
$\frac{\tfrac{3}{2}}5=\tfrac{3}{2}\times\tfrac{1}{5}=\tfrac{3}{10}$
$\frac{8}{\tfrac{1}{3}}=8\times\tfrac{3}{1}=24.$

If, in a complex fraction, there is no clear way to tell which fraction lines takes precedence, then the expression is improperly formed, and ambiguous. Thus 5/10/20/40 is a poorly constructed mathematical expression, with multiple possible values.

### Compound fraction

A compound fraction is a fraction of a fraction, or any number of fractions connected with the word of, corresponding to multiplication of fractions. To reduce a compound fraction to a simple fraction, just carry out the multiplication (see the section on multiplication). For example, $\tfrac{3}{4}$ of $\tfrac{5}{7}$ is a compound fraction, corresponding to $\tfrac{3}{4} \times \tfrac{5}{7} = \tfrac{15}{28}$. The terms compound fraction and complex fraction are closely related and sometimes one is used as a synonym for the other.

### Decimal fractions and percentages

A decimal fractions is a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of digit to the right of a decimal seperator, the appearance of which (e.g., a period, a raised period (•), a comma) depends on the locale. Thus for 0.75 the numerator is 75 and the implied denominator is 10 to the second power, viz.100, because there are two digits to the right of the decimal separator. In decimal numbers greater than 1 (such as 3.75), the fractional part of the number is expressed by the digits to the right of the decimal (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375/100, or as a mixed number, $3\tfrac{75}{100}$.

Decimal fractions can also be expressed using scientific notation with negative exponents, such as 6.023×10−7, which represents 0.0000006023. The 10−7 represents a denominator of 107. Dividing by 107 moves the decimal point 7 places to the left.

Decimal fractions with infinitely many digits to the right of the decimal separator represent an infinite series. For example, ⅓ = 0.333… represents the infinite series 3/10 + 3/100 + 3/1000 + … .