Rational Numbers - Rational Numbers between two Rational Numbers

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, this includes all points on a number line that represent terminating or repeating decimals, situated between the whole integers.

The Density Property of rational numbers states that between any two distinct rational numbers, there are infinitely many other rational numbers. Unlike counting numbers where 3 follows 2, there is no 'next' rational number; you can always zoom in on a number line to find more points between any two values.

To find rational numbers between two fractions with different denominators, the first step is to make the denominators 'like' by finding their Least Common Multiple (LCM). This is like dividing the segments on a number line into equal, smaller increments so they can be easily compared.

The Mean Method is a reliable way to find a rational number exactly in the middle of two others. By calculating $\frac{a + b}{2}$, you find the arithmetic average. On a number line, this result is the precise geometric midpoint between the points $a$ and $b$.

When using equivalent fractions, if there are not enough integers between the numerators to satisfy the requirement (e.g., finding 10 numbers between $\frac{2}{5}$ and $\frac{3}{5}$), multiply both the numerator and denominator of both fractions by a larger number like 10 or $n+1$. This visually expands the 'gap' between the two points without changing their actual value.

Comparing rational numbers on a number line involves looking at their position relative to zero. Positive rational numbers lie to the right of zero, and negative rational numbers lie to the left. When finding numbers between a negative and a positive rational number, zero is often the easiest intermediate number to identify.