A fraction represents a part of of a whole or, more generally, any number of equal parts. A common (or simple) fractions consists of a numerator and a non-zero denominator, separated by a horizontal line (called the fraction bar), such as $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator. Sometimes, the horizontal line is replaced by a slash, such as $a / b$, or by $\div$ such as $a \div b$.

A proper fraction is a fraction where the absolute value of the numerator is less than the absolute value of the denominator, such as $\frac{1}{2}$, $\frac{3}{4}$, etc. An improper fraction is a fraction where the absolute value of the numerator is greater than or equal to the absolute value of the denominator, such as $\frac{5}{4}$, $\frac{7}{2}$, etc. The reciprocal of a fraction $\frac{a}{b}$ is the fraction $\frac{b}{a}$. The reciprocal of a proper fraction is an improper fraction, and vice versa.

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. Multiplying a fraction by its reciprocal results in 1, such as $\frac{a}{b} \cdot \frac{b}{a} = 1$. This is a useful property of fractions.

Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction. If the numerato and denominator are divided by the greatest common divisor of the numerator and the denominator, one gets the equivalent fraction whose numerator and denominator have the lowest absolute values. One says that the fraction has been reduced to its lowest terms. If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be irreducible, reduced, or in simplest terms.

One can add, subtract, multiply and divide two fractions.

1. Addition. If the fractions have the same denominator
   $$\begin{equation} \frac{a}{b}+\frac{c}{b} = \frac{a+c}{b}\end{equation}$$ If the fractions do not have the same denominator it is necessary to bring the two fractions to the same denominator first
   $$\begin{equation} \frac{a}{b}+\frac{c}{d} = \frac{a\cdot d}{b\cdot d} + \frac{c\cdot b}{d\cdot b} = \frac{a\cdot d+c\cdot b}{b\cdot d}\end{equation}$$ For instance, $\frac{3}{4}+\frac{7}{5} = \frac{15}{20} + \frac{28}{20} = \frac{43}{20}$, $\frac{1}{6}+\frac{3}{8} = \frac{4}{24} + \frac{9}{24} = \frac{13}{24}$.
2. Subtraction. Subtracting fractions is, in essence, similar to adding them. If the fractions have the same denominator
   $$\begin{equation} \frac{a}{b}-\frac{c}{b} = \frac{a-c}{b}\end{equation}$$ If the fractions do not have the same denominator it is necessary to bring the two fractions to the same denominator first
   $$\begin{equation} \frac{a}{b}-\frac{c}{d} = \frac{a\cdot d}{b\cdot d} - \frac{c\cdot b}{d\cdot b} = \frac{a\cdot d-c\cdot b}{b\cdot d}\end{equation}$$ For instance, $\frac{3}{4}-\frac{7}{5} = \frac{15}{20} - \frac{28}{20} = -\frac{13}{20}$, $\frac{1}{6}-\frac{3}{8} = \frac{4}{24} - \frac{9}{24} = -\frac{5}{24}$.
3. Multiplication. To multiply fractions, multiply the numerators and multiply the denominators.
   $$\begin{equation} \frac{a}{b} \cdot \frac{c}{d} = \frac{a\cdot c}{b\cdot d}\end{equation}$$ To multiply a faction by a whole number, you can write the whole number as the itseft divided by $1$, then perform normal fraction multiplication.
4. Division. To divide a number by a fraction, multiply that number by the reciprocal of that fraction.
   $$\begin{equation} \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a\cdot d}{b\cdot c}\end{equation}$$ To divide a faction by a whole number, you can write the whole number as the itseft divided by $1$, then perform normal fraction division.

To raise a fraction to a power we raise both the numerator and denominator to that power:

$$\begin{equation}\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}\end{equation}$$ $$\begin{equation}\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}\end{equation}$$