Rational Numbers

1. Definition of Rational Numbers

A rational number is any number that can be expressed as a fraction or quotient \n\frac{a}{b}\n of two integers, where the numerator is $a$ and the non-zero denominator is $b$.

• Formal Notation: $\mathbb{Q} = \left\{\frac{a}{b} \mid a, b \in \mathbb{Z}, b \neq 0\right\}$

• Examples of Rational Numbers:

  • Integers: $-5 = \frac{-5}{1}$, $0 = \frac{0}{1}$, $7 = \frac{7}{1}$

  • Proper and Improper Fractions: $\frac{3}{4}$, $-\frac{11}{5}$

  • Mixed Numbers: $2\frac{1}{3} = \frac{7}{3}$

  • Terminating Decimals: $0.25 = \frac{1}{4}$, $-1.5 = -\frac{3}{2}$

  • Repeating Decimals: $0.333... = \frac{1}{3}$, $0.142857142857... = \frac{1}{7}$

2. Key Properties of Rational Numbers

• Closure Property: Rational numbers are closed under addition, subtraction, and multiplication. Dividing two rational numbers (except by $0$) always results in another rational number.

• Commutative Property:

  • Addition: $a + b = b + a$

  • Multiplication: $a \times b = b \times a$

• Associative Property:

  • Addition: $(a + b) + c = a + (b + c)$

  • Multiplication: $(a \times b) \times c = a \times (b \times c)$

• Distributive Property: $a \times (b + c) = (a \times b) + (a \times c)$

• Identities and Inverses:

  • Additive Identity: $0$ (since $a + 0 = a$)

  • Multiplicative Identity: $1$ (since $a \times 1 = a$)

  • Additive Inverse: $-a$ (since $a + (-a) = 0$)

  • Multiplicative Inverse (Reciprocal): $\frac{1}{a}$ for $a \neq 0$ (since $a \times \frac{1}{a} = 1$)

3. Operations on Rational Numbers

• Addition and Subtraction: Find a common denominator before combining numerators.

  • $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$

  • $\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c}{b \times d}$

• Multiplication: Multiply numerators together and denominators together.

  • $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

• Division: Multiply by the reciprocal of the divisor.

  • $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$

4. Decimal Representation

• Terminating Decimals: A rational number $\frac{a}{b}$ in simplest form converts to a terminating decimal if and only if the prime factors of the denominator $b$ consist only of $2$ and/or $5$.

  • Example: $\frac{3}{8} = 0.375$ (denominator prime factors: $2^3$)

• Repeating Decimals: If the denominator $b$ contains any prime factor other than $2$ or $5$, the decimal expansion will repeat infinitely.

  • Example: $\frac{5}{6} = 0.8333...$ (denominator prime factors: $2 \times 3$)