Understand the historical context and discovery of irrational numbers
Master the geometric proof that √2 is irrational
Learn the complete proof by contradiction for √2's irrationality
Recognize patterns for determining when √n is irrational
Identify famous irrational numbers like π and e
Understand the vast abundance of irrational numbers
9.1 Historical Context
🏛️ Ancient Greeks
Pythagoras and followers believed all numbers were rational
They knew the diagonal of a unit square has length √2
Spent years trying to prove √2 was rational
⚰️ Hippasus (around 500 BCE)
First to prove √2 is irrational
Shocked the Pythagorean community
Legend: Pythagoreans drowned him to suppress this discovery!
💡 The Crisis
The discovery of irrational numbers caused a mathematical crisis in ancient Greece. It shattered the Pythagorean belief that "all is number" (meaning all numbers are ratios of integers).
9.2 Why √2 is Irrational
🔺 Geometric Discovery
1
1
√2
Unit square (sides of length 1)
By Pythagorean theorem: 1² + 1² = c²
Therefore c = √2
But √2 cannot be expressed as p/q!
📋 Proof by Contradiction
Strategy: Assume √2 is rational and derive a contradiction
1Assume √2 = p/q where p and q are integers in lowest terms (gcd(p,q) = 1)
2Square both sides: 2 = p²/q²
3Multiply by q²: 2q² = p²
4Conclusion: p² is even → p is even (only even squares are even)
5Since p is even: Let p = 2k for some integer k
6Substitute: 2q² = (2k)² = 4k² → q² = 2k²
7Conclusion: q² is even → q is even
8CONTRADICTION! Both p and q are even, so they weren't in lowest terms!
The same proof technique (contradiction) can be applied to any √n where n is not a perfect square:
Assume √n = p/q in lowest terms
Square both sides: n = p²/q²
Analyze the prime factorization of n
Show that p and q must share common factors
Contradiction with "lowest terms" assumption
9.4 Famous Irrational Numbers
🥧 π (Pi)
π = 3.1415927...
Ratio of circumference to diameter
Transcendental number
Infinite, non-repeating decimals
📈 e (Euler's number)
e = 2.7182818...
Base of natural logarithm
Also transcendental
Fundamental in calculus
🌟 Golden Ratio (φ)
φ = (1 + √5)/2 ≈ 1.618...
Appears in nature and art
Related to Fibonacci sequence
Algebraic irrational
🔢 Other Examples
log₂(3), √(2), ∛(5), ...
Infinite variety of irrationals
Much more abundant than rationals
Fill the "gaps" in rational numbers
🌊 Properties of Irrational Numbers
Infinite, non-repeating decimal expansions
Cannot be expressed as fractions
There are VASTLY more irrational numbers than rational numbers!
Between any two numbers, there are infinitely many irrationals
🧮 Practice Problems
Problem 1: Identifying Irrationals
Determine which of the following are irrational:
√49
√50
√64
√72
√100
Solutions:
√49 = 7 - RATIONAL (perfect square)
√50 - IRRATIONAL (50 is not a perfect square)
√64 = 8 - RATIONAL (perfect square)
√72 - IRRATIONAL (72 is not a perfect square)
√100 = 10 - RATIONAL (perfect square)
Problem 2: Proof Practice
Prove that √3 is irrational using the same technique as for √2.
Proof by Contradiction:
Assume √3 = p/q in lowest terms
Square: 3 = p²/q² → 3q² = p²
Conclusion: p² is divisible by 3 → p is divisible by 3
Let p = 3k: 3q² = (3k)² = 9k² → q² = 3k²
Conclusion: q² is divisible by 3 → q is divisible by 3
CONTRADICTION: Both p and q divisible by 3, not lowest terms!
∴ √3 is irrational
Problem 3: Decimal Representations
Explain why the following decimal representations indicate irrational numbers:
0.101001000100001000001...
0.1234567891011121314...
3.141592653589793...
Solutions:
Pattern but non-repeating: The pattern of 0s increases, so it never repeats a fixed block
Non-repeating sequence: Contains all natural numbers in order, creating a non-repeating, infinite decimal
π (pi): Known to be transcendental and irrational, with infinite non-repeating decimals
Key insight: Rational numbers have either terminating or eventually repeating decimal expansions. These examples have infinite, non-repeating patterns, so they must be irrational.
Problem 4: Operations with Irrationals
Determine if the following are rational or irrational: