Rational Numbers - Operations on Rational Numbers (Addition, Subtraction, Multiplication, Division)

Rational Numbers are numbers that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Visually, they represent points on a number line that can exist between integers.

To add rational numbers with the same denominator, simply add the numerators and keep the common denominator. For example, $\frac{2}{7} + \frac{3}{7}$ means taking 2 parts and 3 parts of a whole divided into 7 equal sections, resulting in $\frac{5}{7}$.

To add or subtract rational numbers with different denominators, first find the Least Common Multiple (LCM) of the denominators to convert them into like fractions. This is like resizing the sub-divisions on a number line so that the jumps are of equal length.

The Additive Inverse of a rational number $\frac{a}{b}$ is $-\frac{a}{b}$, such that their sum is 0. On a number line, this is the point at the same distance from zero but in the opposite direction.

Subtraction of rational numbers is defined as adding the additive inverse of the number being subtracted. For example, $\frac{2}{3} - \frac{1}{4}$ is the same as $\frac{2}{3} + (-\frac{1}{4})$.

Multiplication involves multiplying the numerators together and the denominators together. If you visualize a rectangle with side lengths $\frac{a}{b}$ and $\frac{c}{d}$, the area represents the product $\frac{ac}{bd}$.

The Reciprocal (Multiplicative Inverse) of a non-zero rational number $\frac{a}{b}$ is $\frac{b}{a}$. Multiplying a number by its reciprocal always results in 1.

Division of one rational number by another is performed by multiplying the first number by the reciprocal of the second. This is expressed as $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.