Rational Numbers

A. The Axiomatic Definition:

A Rational Number is any number that can be expressed in the form:

• $p$ is the numerator (can be positive, negative, or zero).
• $q$ is the denominator (must be a non-zero integer).
• Integers as Rational Numbers: Every integer $n \in \mathbb{Z}$ is a rational number because it can be written with denominator 1: $n = \frac{n}{1}$ (e.g., $-7 = \frac{-7}{1}$, $0 = \frac{0}{1}$).

B. Standard Form of a Rational Number:

A rational number $\frac{p}{q}$ is said to be in Standard Form if and only if:

1. Its denominator $q$ is a positive integer ($q > 0$). (If negative, multiply both numerator and denominator by $-1$: $\frac{5}{-8} = \frac{-5}{8}$).
2. The numerator $p$ and denominator $q$ have no common factor other than 1 (i.e., they are coprime: $\gcd(|p|, q) = 1$).

A. Equivalent Rational Numbers:

For any non-zero integer $m \neq 0$:

Example: Equivalent forms of $\frac{-3}{5}$ are $\frac{-6}{10}, \frac{-9}{15}, \frac{-12}{20}$.

B. Comparison of Two Rational Numbers:

• Method 1: Cross-Multiplication:

  Convert both numbers so their denominators are positive: $\frac{a}{b}$ and $\frac{c}{d}$ (with $b, d > 0$). Compute the cross-products $a \cdot d$ and $b \cdot c$:

  • If $a \cdot d > b \cdot c \implies \frac{a}{b} > \frac{c}{d}$
  • If $a \cdot d = b \cdot c \implies \frac{a}{b} = \frac{c}{d}$
  • If $a \cdot d < b \cdot c \implies \frac{a}{b} < \frac{c}{d}$
• Method 2: Common Denominator (LCM): Express both fractions with the positive $\text{LCM}(b, d)$ and compare their numerators directly.

Unlike integers (where there are zero integers between 4 and 5), rational numbers possess the Density Property: Between any two distinct rational numbers, there exist an infinite number of rational numbers!

Two Systematic Methods to Find $n$ Rational Numbers:

• Method 1: Equivalent Denominator Scaling:

  To find 5 rational numbers between $\frac{1}{3}$ and $\frac{1}{2}$:

  1. Make denominators equal using LCM of 3 and 2 ($= 6$): $\frac{1}{3} = \frac{2}{6}$ and $\frac{1}{2} = \frac{3}{6}$.
  2. Since we need 5 numbers, scale both by $5 + 1 = 6$:
     $$\frac{2 \times 6}{6 \times 6} = \frac{12}{36}, \quad \frac{3 \times 6}{6 \times 6} = \frac{18}{36}$$
  3. The 5 required numbers are: $\mathbf{\frac{13}{36}, \frac{14}{36}, \frac{15}{36}, \frac{16}{36}, \frac{17}{36}}$.
• Method 2: Mean / Average Method: The rational number lying exactly halfway between $x$ and $y$ is $m = \frac{x + y}{2}$.

A. Addition & Subtraction:

B. Multiplication & Division (Reciprocals):

• Multiplication: Multiply numerators together and denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
• Multiplicative Inverse (Reciprocal): The reciprocal of a non-zero rational number $\frac{a}{b}$ is $\frac{b}{a}$ because $\frac{a}{b} \times \frac{b}{a} = 1$. (Note: $0$ has no reciprocal!).
• Division: To divide by a rational number, multiply by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c} \quad (c \neq 0)$$