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

A rational number is defined as a number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Visually, you can imagine a number line where rational numbers are points located between the integers; for example, $\frac{1}{2}$ is exactly halfway between $0$ and $1$.

Addition and subtraction of rational numbers with the same denominator involves simply adding or subtracting the numerators while keeping the denominator constant. Visualize a circle divided into equal parts (like a pizza); if you have $2$ slices of an $8$-slice pizza and add $3$ more slices, you have $\frac{2}{8} + \frac{3}{8} = \frac{5}{8}$ of the pizza.

When adding or subtracting rational numbers with different denominators, you must find the Least Common Multiple (LCM) of the denominators to make them like fractions. On a number line, this is equivalent to finding a common scale or smaller sub-divisions that both fractions can be represented by precisely.

Multiplication of rational numbers is performed by multiplying the numerators together and the denominators together. Visually, multiplying $\frac{1}{2} \times \frac{1}{2}$ can be seen as taking half of a half-shaded square, resulting in a region that covers $\frac{1}{4}$ of the total area.

Division of rational numbers is the process of multiplying the first rational number by the reciprocal of the second. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$. Imagine you are asking 'How many times does $\frac{1}{4}$ fit into $\frac{1}{2}$?'; the visual result is $2$ whole parts.

Every rational number $\frac{a}{b}$ has an additive inverse $-\frac{a}{b}$ such that their sum is $0$. On a number line, these are points located at the same distance from $0$ but in opposite directions (mirror images across the origin).

The number $0$ is the additive identity (adding it doesn't change the number), and $1$ is the multiplicative identity (multiplying by it doesn't change the value).

A rational number is in its standard form if the denominator $q$ is positive and there is no common factor between $p$ and $q$ other than $1$. Visually, this is the most simplified 'name' for a point on the number line.