The formula for rational numbers, expressed as $\frac{p}{q}$, is rooted in the definition and nature of numbers that can be represented as fractions or ratios of integers. Here's why rational numbers are expressed in this way:

1. Definition of Rational Numbers: A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where:

   • $p$ and $q$ are integers (whole numbers).
   • $q \neq 0$ (to avoid division by zero, which is undefined).

   This definition highlights that rational numbers are "ratios" of two integers, which is where the term "rational" comes from.

2. Division as a Fundamental Operation: Division is a way to break down one quantity relative to another. The fraction $\frac{p}{q}$ represents how many parts of size $\frac{1}{q}$ are needed to make $p$. This ratio is the essence of rational numbers.

3. Practical Examples: Rational numbers naturally arise in everyday contexts where quantities are divided. For example:

   • If you cut a pizza into 4 pieces ($q = 4$) and take 3 pieces ($p = 3$), you have $\frac{3}{4}$ of the pizza.
   • Similarly, 0.5 can be expressed as $\frac{1}{2}$, showing that it is a rational number.

4. Connection to Real Numbers: Rational numbers are part of the real numbers and provide a way to represent values precisely using fractions. Some real numbers, such as $\pi$ or $\sqrt{2}$, cannot be expressed as a ratio of integers and are called irrational.

5. Mathematical Representation: Using $\frac{p}{q}$:

   • Ensures every rational number has a clear and unambiguous representation.
   • Allows for easy operations like addition, subtraction, multiplication, and division using fraction rules.

Why Is the Formula $\frac{p}{q}$ Specifically Chosen?

This choice is not arbitrary; it is based on the concept of numbers that repeat or terminate when written as decimals (e.g., $0.5$ or $0.\overline{333}$). Any such number can always be converted into a fraction $\frac{p}{q}$.

Questions to Explore:

1. Why can all terminating or repeating decimals be represented as rational numbers?
2. How does the concept of rational numbers differ from irrational numbers?
3. Why must $q \neq 0$ in the definition of rational numbers?
4. Can a whole number like $5$ also be considered a rational number?
5. How does the concept of ratios relate to the real-world application of fractions?

Tip:

Understanding rational numbers as ratios helps in visualizing fractions and their operations. Think of $\frac{p}{q}$ as splitting something into $q$ parts and taking $p$ of those parts.