Understanding Rational Numbers

Rational Numbers Revision Sheet

1. 📌 Essentials

• Rational numbers are numbers that can be written as a fraction $\frac{p}{q}$ with integers $p, q$ and $q \neq 0$.
• They include integers (when denominator 1).
• Simplification involves dividing numerator and denominator by their GCD.
• Two fractions are equivalent if cross-multiplied: $p \times s = r \times q$.
• Operations follow standard fraction rules: addition, subtraction, multiplication, division.
• Rational numbers are dense: between any two rationals, another rational exists.
• They are countable subsets of real numbers.
• Rational numbers can be positive, negative, or zero.
• The set of rational numbers is denoted as $\mathbb{Q}$.
• Rational numbers are crucial for precise ratios and divisions.

2. 🧩 Key Structures & Components

• Numerator ($p$) — top part of the fraction, represents the part or numerator.
• Denominator ($q$) — bottom part, must be non-zero, indicates the division.
• GCD (Greatest Common Divisor) — used to simplify fractions.
• Equivalent fractions — different fractions representing the same value.
• Operations:
  • Addition: $\frac{p}{q} + \frac{r}{s}$
  • Subtraction: $\frac{p}{q} - \frac{r}{s}$
  • Multiplication: $\frac{p}{q} \times \frac{r}{s}$
  • Division: $\frac{p}{q} \div \frac{r}{s}$

3. 🔬 Functions, Mechanisms & Relationships