Topic 2: Rational Numbers | Mathematics for Data Science

🎯 Learning Objectives

Understand the definition and notation of rational numbers (ℚ)
Learn about non-unique representation of rational numbers
Master the concept of reduced form and greatest common divisor
Explore the density property of rational numbers
Understand why rational numbers have no "next" number

2.1 Definition

📝 Notation

ℚ (from "quotient" or "ratio")

📚 Definition

Rational Number: Any number that can be expressed as p/q where:

p, q ∈ ℤ (both are integers)
q ≠ 0 (denominator cannot be zero)

💡 Examples of Rational Numbers

Positive Fractions

3/5, 7/8, 22/7

Negative Fractions

-7/4, -15/23, -1/3

Integers as Rationals

5 = 5/1, -3 = -3/1, 0 = 0/1

Terminating Decimals

0.75 = 3/4, 0.125 = 1/8

🔗 Relationship to Integers

Every integer is a rational number because any integer n can be written as n/1. Therefore: ℤ ⊆ ℚ

2.2 Non-Unique Representation

🔑 Key Property

Rational numbers do NOT have unique representations

📊 Demonstration

3/5

6/10

9/15

30/50

-15/-25

All these fractions represent the same rational number!

🤔 Why This Happens

Multiplying both numerator and denominator by the same non-zero number gives an equivalent fraction:

$$\frac{p}{q} = \frac{p \times k}{q \times k} \text{ for any } k \neq 0$$

🎮 Interactive Example

Let's see how 2/3 can be represented in multiple ways:

2/3 × 2/2 = 4/6

2/3 × 5/5 = 10/15

2/3 × (-3)/(-3) = -6/-9

2.3 Reduced Form (Canonical Form)

📚 Definition

A rational number p/q is in reduced form when:

gcd(p, q) = 1 (p and q have no common factors other than 1)
This gives a unique representation

💡 Example: Reducing 18/60

18/60

→

Find gcd(18, 60)

→

gcd = 6

→

18÷6 / 60÷6

→

3/10

18/60 = 3/10 (in reduced form)

✨ Uniqueness

Once in reduced form, every rational number has exactly one representation (ignoring the sign placement).

2.4 Greatest Common Divisor (GCD)

📚 Definition

gcd(m, n) is the largest number that divides both m and n

Finding GCD using Prime Factorization

Find prime factorization of both numbers

Identify common prime factors

Multiply the common primes

🔍 Detailed Example: gcd(18, 60)

2 × 3 × 3

= 2¹ × 3²

2 × 2 × 3 × 5

= 2² × 3¹ × 5¹

Common Prime Factors:

2: min(1, 2) = 1, so we take 2¹
3: min(2, 1) = 1, so we take 3¹
5: min(0, 1) = 0, so we don't include 5

gcd(18, 60) = 2¹ × 3¹ = 6

📝 Note

There are more efficient algorithms (like the Euclidean algorithm) to find GCD, but prime factorization provides intuitive understanding.

2.5 Density of Rational Numbers

🏛️ Theorem

Rational numbers are dense

📚 Definition of Density

Between any two rational numbers, there exists another rational number

🔍 Proof Sketch

For any r < r', the average (r + r')/2 lies between them

(r + r')/2

🎯 Implication

There is NO "next" rational number - unlike integers, we cannot find the immediate successor!

💡 Example: Between 1/3 and 1/2

Step 1: Find the average

$$\frac{\frac{1}{3} + \frac{1}{2}}{2} = \frac{\frac{2+3}{6}}{2} = \frac{5/6}{2} = \frac{5}{12}$$

So 5/12 lies between 1/3 and 1/2

Step 2: We can continue infinitely!

Between 1/3 and 5/12: average = (1/3 + 5/12)/2 = 3/8
Between 5/12 and 1/2: average = (5/12 + 1/2)/2 = 11/24
Between 3/8 and 5/12: and so on...

🔄 Infinite Process

This process can continue forever, showing that there are infinitely many rational numbers between any two rationals!

👁️ Visual Representation

1/3

3/8

5/12

11/24

1/2

New rational numbers keep appearing between existing ones!

🏃‍♂️ Practice Problems

Problem 1: Reducing to Lowest Terms

Reduce the following fractions to their lowest terms:

(a) 24/36 (b) 45/75 (c) 48/64

Problem 2: Finding GCD

Find gcd(120, 180) using prime factorization.

Problem 3: Density Property

Find a rational number between 2/5 and 3/7.

🔑 Key Takeaways

Rational numbers are fractions p/q where p, q ∈ ℤ and q ≠ 0
Non-unique representation: Same rational can be written in infinitely many ways
Reduced form: Unique representation when gcd(p, q) = 1
GCD: Found using prime factorization or Euclidean algorithm
Density: Between any two rationals, there's always another rational
No "next" rational: Unlike integers, rationals have no immediate successor

🔢 Topic 2: Rational Numbers

🎯 Learning Objectives

2.1 Definition

📝 Notation

📚 Definition

💡 Examples of Rational Numbers

Positive Fractions

Negative Fractions

Integers as Rationals

Terminating Decimals

🔗 Relationship to Integers

2.2 Non-Unique Representation

🔑 Key Property

📊 Demonstration

🤔 Why This Happens

🎮 Interactive Example

2.3 Reduced Form (Canonical Form)

📚 Definition

💡 Example: Reducing 18/60

✨ Uniqueness

2.4 Greatest Common Divisor (GCD)

📚 Definition

Finding GCD using Prime Factorization

Find prime factorization of both numbers

Identify common prime factors

Multiply the common primes

🔍 Detailed Example: gcd(18, 60)

Common Prime Factors:

📝 Note

2.5 Density of Rational Numbers

🏛️ Theorem

📚 Definition of Density

🔍 Proof Sketch

🎯 Implication

💡 Example: Between 1/3 and 1/2

Step 1: Find the average

Step 2: We can continue infinitely!

🔄 Infinite Process

👁️ Visual Representation

🏃‍♂️ Practice Problems

Problem 1: Reducing to Lowest Terms

Solutions:

Problem 2: Finding GCD

Solution:

Problem 3: Density Property

Solution:

🔑 Key Takeaways