Rational Numbers

Integers Revision

Theory

Before we dive into rational numbers, let’s take a moment to review integers and why we need to expand our number system even further. Here’s why this matters: while integers gave us the ability to subtract any two whole numbers, they still can’t handle all division problems.

When we divide integers, we often get results that aren’t integers themselves:

\[8 \div 2 = 4 \quad \text{(This works perfectly in integers)}\]

\[7 \div 2 = ? \quad \text{(This doesn't give us a whole number)}\]

This limitation is exactly what led mathematicians to develop rational numbers - a brilliant solution that allows us to express any division result precisely!

Key Integer Properties We’ll Build Upon:

• Closure under multiplication: \(a \times b \in \mathbb{Z}\) for all integers \(a, b\) • Distributive property: \(a(b + c) = ab + ac\) • Commutative property: \(a + b = b + a\) and \(a \times b = b \times a\) • Associative property: \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\)

Notice how these properties will help us understand fraction operations later!

Application

Examples

Example 1: When Integer Division Works Perfectly

Let’s explore cases where division stays within integers:

\[(-24) \div 6 = -4 \quad \text{(Negative divided by positive gives negative)}\]

\[36 \div (-9) = -4 \quad \text{(Positive divided by negative gives negative)}\]

\[(-45) \div (-5) = 9 \quad \text{(Negative divided by negative gives positive)}\]

These examples show that integers are sometimes sufficient for division, but what about when they’re not?

Example 2: The Need for Rational Numbers

Here’s a typical problem that shows why we need fractions:

\[13 \div 4 = 3 \text{ remainder } 1 \quad \text{(Integer division with remainder)}\]

But this doesn’t tell us the complete story! The exact answer is:

\[13 \div 4 = \frac{13}{4} = 3\frac{1}{4} = 3.25 \quad \text{(Rational number representation)}\]

Interactive Visualization: From Integers to Fractions

Interactive Graph

Integer division and fraction visualization will be implemented here

Multiple Choice Questions

Rational Numbers

Theory

Let’s explore the ingenious solution to our division problem: rational numbers! A rational number is any number that can be expressed as a fraction of two integers, where the denominator isn’t zero.

The Set of Rational Numbers:

\[\mathbb{Q} = \left\{\frac{a}{b} : a, b \in \mathbb{Z}, b \neq 0\right\}\]

Here’s why this definition is so powerful: it allows us to represent any division result exactly, not just as an approximation!

Understanding Fraction Terminology:

Parts of a Fraction: • Numerator (top): The dividend - what’s being divided • Denominator (bottom): The divisor - what we’re dividing by • Fraction bar: Represents the division operation

Types of Fractions:

Proper Fractions: When the numerator is smaller than the denominator

\[\frac{3}{5}, \quad \frac{2}{7}, \quad \frac{11}{15} \quad \text{(All less than 1)}\]

Improper Fractions: When the numerator is greater than or equal to the denominator

\[\frac{7}{3}, \quad \frac{9}{4}, \quad \frac{15}{15} \quad \text{(Greater than or equal to 1)}\]

Mixed Numbers: A whole number combined with a proper fraction

\[2\frac{1}{3} = \frac{7}{3}, \quad 5\frac{3}{4} = \frac{23}{4}\]

Equivalent Fractions - The Key to Understanding Rational Numbers:

Here’s a crucial insight: many different fractions represent the same rational number!

\[\frac{a}{b} = \frac{a \times k}{b \times k} \quad \text{for any non-zero integer } k\]

For example: $\(\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = \frac{50}{100}\)$

This is why we often simplify fractions to their lowest terms by dividing both numerator and denominator by their greatest common divisor (GCD).

Fundamental Properties of Rational Numbers:

Density Property: Between any two rational numbers, there’s always another rational number

\[\text{Between } \frac{1}{3} \text{ and } \frac{1}{2}, \text{ we have } \frac{5}{12} \text{ (and infinitely many others!)}\]

Closure Properties: Rational numbers are closed under all four basic operations (except division by zero)

• Addition: \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\) • Subtraction: \(\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\) • Multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\) • Division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\) (when \(c \neq 0\))

Relationship to Other Number Systems:

Notice how rational numbers include all our previous number systems: • Every natural number \(n\) is rational: \(n = \frac{n}{1}\) • Every integer \(m\) is rational: \(m = \frac{m}{1}\) • This means: \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}\)

Decimal Representations:

It’s worth taking a moment to appreciate that every rational number has a decimal representation that either: • Terminates: \(\frac{3}{4} = 0.75\) • Repeats: \(\frac{1}{3} = 0.333...\) or \(\frac{2}{7} = 0.285714285714...\)

This property distinguishes rational numbers from irrational numbers!

Interactive Visualization: Rational Numbers on the Number Line

Interactive Graph

Rational number placement and equivalence visualization will be implemented here

Application

Examples

Example 1: Simplifying Fractions

Let’s work through simplifying \(\frac{48}{72}\). This might look complex at first, but we’ll break it down systematically.

Method 1: Finding the GCD

Here’s how we approach this using prime factorization:

\[48 = 2^4 \times 3 = 16 \times 3 \quad \text{(Factor 48)}\]

\[72 = 2^3 \times 3^2 = 8 \times 9 \quad \text{(Factor 72)}\]

\[\text{GCD}(48, 72) = 2^3 \times 3 = 24 \quad \text{(Take lowest powers of common factors)}\]

\[\frac{48}{72} = \frac{48 \div 24}{72 \div 24} = \frac{2}{3} \quad \text{(Divide by GCD)}\]

Method 2: Step-by-Step Simplification

Notice what happens when we divide by common factors one at a time:

\[\frac{48}{72} = \frac{48 \div 2}{72 \div 2} = \frac{24}{36} \quad \text{(Both even, divide by 2)}\]

\[\frac{24}{36} = \frac{24 \div 2}{36 \div 2} = \frac{12}{18} \quad \text{(Still even, divide by 2 again)}\]

\[\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3} \quad \text{(Divide by 6)}\]

Example 2: Adding Fractions with Different Denominators

Let’s add \(\frac{3}{8} + \frac{5}{12}\). The key insight here is finding a common denominator.

Finding the Least Common Denominator (LCD):

\[8 = 2^3, \quad 12 = 2^2 \times 3 \quad \text{(Prime factorization)}\]

\[\text{LCD} = 2^3 \times 3 = 24 \quad \text{(Take highest powers)}\]

Converting to Common Denominator:

\[\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \quad \text{(Multiply by } \frac{3}{3} = 1\text{)}\]

\[\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24} \quad \text{(Multiply by } \frac{2}{2} = 1\text{)}\]

Adding the Fractions:

\[\frac{9}{24} + \frac{10}{24} = \frac{19}{24} \quad \text{(Add numerators, keep denominator)}\]

Example 3: Converting Mixed Numbers

A recipe calls for \(3\frac{2}{5}\) cups of flour. Convert this to an improper fraction.

Here’s a helpful way to think about this conversion:

\[3\frac{2}{5} = 3 + \frac{2}{5} \quad \text{(Separate whole and fractional parts)}\]

\[= \frac{3 \times 5}{5} + \frac{2}{5} \quad \text{(Express 3 as fifths)}\]

\[= \frac{15}{5} + \frac{2}{5} = \frac{17}{5} \quad \text{(Combine the fractions)}\]

Quick Method: \((3 \times 5) + 2 = 17\), so \(3\frac{2}{5} = \frac{17}{5}\)

Multiple Choice Questions

Sector Specific Questions: Rational Numbers Applications

Key Takeaways

Important

1. Rational numbers (\(\mathbb{Q}\)) are numbers that can be expressed as \(\frac{a}{b}\) where \(a, b \in \mathbb{Z}\) and \(b \neq 0\)

2. Every integer is rational: Any integer \(n\) can be written as \(\frac{n}{1}\)

3. Equivalent fractions represent the same value: \(\frac{2}{3} = \frac{4}{6} = \frac{6}{9}\)

4. Simplification means dividing numerator and denominator by their GCD

5. Operations require common denominators for addition and subtraction

6. Multiplication is straightforward: Multiply numerators and denominators separately

7. Division means multiply by the reciprocal: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

8. Rational numbers are dense: Between any two rationals, there’s always another rational