Rational Numbers

A. Definition & Standard Form:

A number that can be expressed in the form $\mathbf{\frac{p}{q}}$, where $p$ and $q$ are integers and $\mathbf{q \ne 0}$, is called a Rational Number ($\mathbb{Q}$).

• Standard Form: A rational number $\frac{p}{q}$ is in standard form if:
  1. Its denominator $q$ is strictly positive ($q > 0$).
  2. The numerator $p$ and denominator $q$ have no common factor other than $1$ (they are co-prime: $\gcd(|p|, q) = 1$).
  3. Example: Express $\frac{36}{-54}$ in standard form: $\frac{36 \div (-18)}{-54 \div (-18)} = \mathbf{-\frac{2}{3}}$.

B. The Density Property of Rational Numbers:

Between any two distinct rational numbers $a$ and $b$, there exist infinitely many rational numbers! There is no such thing as the "next" rational number.

1. Closure Property: Rational numbers are closed under Addition, Subtraction, and Multiplication. $$a + b \in \mathbb{Q}, \quad a - b \in \mathbb{Q}, \quad a \times b \in \mathbb{Q}$$ *(Division is closed ONLY if the divisor is non-zero: $\frac{a}{b} \in \mathbb{Q}$ for $b \ne 0$)*.
2. Commutative Property:
   • Addition is commutative: $a + b = b + a$.
   • Multiplication is commutative: $a \times b = b \times a$.
   • *Subtraction and Division are NOT commutative!* ($a - b \ne b - a$ and $a \div b \ne b \div a$).
3. Associative Property: $$(a + b) + c = a + (b + c) \quad \land \quad (a \times b) \times c = a \times (b \times c)$$
4. Distributive Property of Multiplication over Addition/Subtraction: $$\mathbf{a(b \pm c) = ab \pm ac}$$
5. Identities & Inverses:
   • Additive Identity: $0$ ($a + 0 = 0 + a = a$).
   • Multiplicative Identity: $1$ ($a \times 1 = 1 \times a = a$).
   • Additive Inverse: $-a$ (such that $a + (-a) = 0$).
   • Multiplicative Inverse (Reciprocal): $\frac{1}{a}$ for $a \ne 0$ (such that $a \times \frac{1}{a} = 1$). Zero has NO reciprocal!

Method 1: The Mean / Average Method:

To find a rational number strictly between $a$ and $b$ (where $a < b$):

Repeat successively to generate multiple numbers: $a < \frac{a + m}{2} < m < \frac{m + b}{2} < b$.

Method 2: Common Denominator & Scaling Method (Faster for multiple numbers):

To insert $n$ rational numbers between $\frac{a}{b}$ and $\frac{c}{d}$:

1. Convert both fractions to have an identical common denominator using LCM.
2. Multiply both numerator and denominator of both fractions by $(n + 1)$ (or 10).
3. Select intermediate numerators directly!
4. Example: Insert 5 rational numbers between $-\frac{1}{3}$ and $\frac{1}{2}$:

   Common denominator LCM(3, 2) = 6: $-\frac{2}{6}$ and $\frac{3}{6}$. Multiply by 2: $-\frac{4}{12}$ and $\frac{6}{12}$.

   Intermediate rational numbers: $\mathbf{-\frac{3}{12}, -\frac{2}{12}, -\frac{1}{12}, \frac{1}{12}, \frac{2}{12}}$.

The Prime Factorisation Test:

Let $x = \frac{p}{q}$ be a rational number in its simplest standard form ($\gcd(p, q) = 1$):

• Terminating Decimal: The decimal expansion terminates (ends) if and only if the prime factorisation of the denominator $q$ contains ONLY powers of 2 and/or 5: $$\mathbf{q = 2^m \times 5^n \quad (m, n \in \mathbb{W})}$$ Examples: $\frac{7}{40} = \frac{7}{2^3 \times 5^1} = 0.175$ (terminates!). $\frac{3}{25} = \frac{3}{5^2} = 0.12$ (terminates!).
• Non-Terminating Repeating (Recurring) Decimal: If $q$ has any prime factor other than 2 or 5 (such as 3, 7, 11, 13), the decimal expansion is non-terminating and periodic: Examples: $\frac{1}{3} = 0.333\dots = 0.\bar{3}$; $\quad \frac{2}{7} = 0.\overline{285714}$.