Are rational numbers enough to represent every point on the number line?
Consider square roots:
What about √2, √3, √5, √6, √7, √8, ...?
These numbers exist (they have a position on the number line), but are they rational?
By Pythagorean Theorem:
Therefore: diagonal = √2
This length clearly exists, but is √2 a rational number?
Real Numbers (ℝ): The union of all rational and irrational numbers
Every point on the number line corresponds to a unique real number
Real numbers completely fill the number line with no gaps
All points on number line
p/q form
Not expressible as p/q
..., -2, -1, 0, 1, 2, ...
0, 1, 2, 3, ...
Rational: Can be expressed as a fraction p/q (where p, q ∈ ℤ, q ≠ 0)
Irrational: Cannot be expressed as a fraction - decimal expansion goes on forever without repeating
Pythagoras and his followers believed that "All is number"
By "number," they meant rational numbers
They thought every length could be expressed as a ratio of integers
Pythagoreans knew the diagonal of a unit square had length √2
They spent years trying to prove √2 was rational
Problem: They couldn't find integers p and q such that p/q = √2
Hippasus proved that √2 is irrational
This shattered the Pythagorean worldview!
Legend: The Pythagoreans were so disturbed by this discovery that they allegedly drowned Hippasus at sea to suppress the truth
The discovery of irrational numbers was a crisis in ancient mathematics
It showed that reality was more complex than previously thought
It forced mathematicians to develop more sophisticated number systems
√2 is irrational
i.e., there do not exist integers p and q such that √2 = p/q
We'll assume √2 IS rational, and show this leads to a logical impossibility
Assume √2 is rational
Then we can write: √2 = p/q where p, q ∈ ℤ, q ≠ 0
Important: We assume p/q is in lowest terms (i.e., gcd(p, q) = 1)
From p² = 2q², we see that p² is even
Fact: If p² is even, then p must be even
(Because the square of an odd number is always odd)
Therefore, we can write: p = 2k for some integer k
From 2k² = q², we see that q² is even
Therefore, q must be even too
We've shown that both p and q are even
This means they share a common factor of 2
But we assumed p/q was in lowest terms (gcd(p, q) = 1)!
This is a CONTRADICTION
Our assumption must be wrong
Therefore, √2 cannot be written as p/q
√2 is irrational ∎
The proof cleverly uses the uniqueness of prime factorization
When we square both sides, we create an equation where the left side has an odd power of 2, but the right side has an even power of 2
This is impossible if prime factorization is unique!
Diagonal of unit square
First known irrational number
Height of equilateral triangle with side 2
Proven irrational by similar method
Ratio of circumference to diameter
Appears throughout mathematics
Proven irrational in 1761
Base of natural logarithm
e = 1 + 1/1! + 1/2! + 1/3! + ...
Central to calculus
φ = (1 + √5)/2
Appears in art and nature
Theorem: √n is irrational
whenever n is not a perfect square
Theorem: If n is a positive integer that is not a perfect square, then √n is irrational
Proof: Similar to the proof for √2, using contradiction and prime factorization
Irrational numbers have infinite, non-repeating decimal expansions
√2 = 1.41421356237309504880168872420969807856967...
π = 3.14159265358979323846264338327950288419716...
These go on forever without ever repeating a pattern!
Terminating or repeating decimals
1/4 = 0.25 (terminates)
1/3 = 0.333... (repeats)
22/7 = 3.142857142857... (repeats)
Non-terminating, non-repeating decimals
√2 = 1.414213... (never repeats)
π = 3.141592... (never repeats)
e = 2.718281... (never repeats)
Example: 1 + √2 is irrational
√2 + √2 = 2√2 (irrational)
√2 + (-√2) = 0 (rational)
Like rationals, irrational numbers are also dense
Between any two real numbers, there exist:
Surprising fact: Despite being dense, rational numbers are "countable" while irrational numbers are "uncountable" (covered in later topics)
Given a decimal expansion
Does it terminate?
RATIONAL
Examples: 0.5, 0.75, 3.125
Does it repeat?
RATIONAL
Examples: 0.333..., 0.142857142857...
IRRATIONAL
Examples: √2, π, e
No pattern ever repeats
Goes on forever without repeating
Classify each of the following as rational or irrational:
(a) √16 (b) √7 (c) 0.125 (d) 0.121212... (e) π/2
Using a proof similar to √2, prove that √3 is irrational.
Which of the following decimal expansions represent rational numbers?
(a) 0.123123123... (b) 0.123456789101112... (c) 2.5000... (d) 0.1011011101111...
Determine whether each result is rational or irrational:
(a) √2 + 3 (b) √2 × √8 (c) √5 + (-√5) (d) 2√3
A square has diagonal length 10. What is the side length? Is it rational or irrational?