Pythagorean triples and irrational numbers are key concepts in ancient Greek mathematics. These ideas challenged the Pythagorean belief that all was number and led to a crisis in Greek math.

The discovery of irrational numbers, like √2, expanded our understanding of numbers beyond rationals. This breakthrough paved the way for new mathematical concepts and methods of proof.

Pythagorean Triples

Understanding Pythagorean Triples and Their Properties

• Pythagorean triples consist of three positive integers (a, b, c) satisfying the equation $a^2 + b^2 = c^2$
• Represent the side lengths of right triangles where c is the hypotenuse
• Common examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17)
• Can be generated using the formulas: $a = m^2 - n^2, b = 2mn, c = m^2 + n^2$ where m and n are positive integers with m > n
• Primitive Pythagorean triples have no common factors among the three numbers
• (3, 4, 5) is a primitive triple, while (6, 8, 10) is not primitive as it's a multiple of (3, 4, 5)
• To generate primitive triples, m and n must be coprime and not both odd

Rational Numbers and Their Relationship to Pythagorean Triples

• Rational numbers express as fractions $\frac{p}{q}$ where p and q are integers and q ≠ 0
• All Pythagorean triples consist of rational numbers
• Ratios of Pythagorean triple components always yield rational numbers
• Pythagorean triples can be used to approximate irrational numbers (√2 ≈ 7/5)
• Rational solutions to the Pythagorean equation correspond to points with rational coordinates on the unit circle

Irrational Numbers

Defining and Exploring Irrational Numbers

• Irrational numbers cannot express as fractions $\frac{p}{q}$ where p and q are integers and q ≠ 0
• Have non-repeating, non-terminating decimal representations
• Include famous constants like π, e, and √2
• Discovered by the Pythagoreans when studying the diagonal of a unit square
• Square root of 2 (√2) serves as a classic example of an irrational number
• √2 approximately equals 1.41421356..., with digits continuing infinitely without pattern

Incommensurable Lengths and Their Significance

• Incommensurable lengths lack a common unit of measurement
• Diagonal and side of a square exemplify incommensurable lengths
• Led to a crisis in Greek mathematics, challenging the Pythagorean belief that all was number
• Expanded the concept of number beyond rational numbers
• Resulted in the development of geometric algebra to handle irrational magnitudes

Algebraic and Transcendental Numbers

• Algebraic numbers serve as roots of polynomial equations with integer coefficients
• Include all rational numbers and some irrational numbers (√2, ³√5)
• Transcendental numbers are irrational numbers that are not algebraic
• π and e are famous examples of transcendental numbers
• Transcendental numbers are "more irrational" than algebraic irrationals
• Proved to exist by Liouville in 1844, with π proven transcendental by Lindemann in 1882

Proving Irrationality

Proof by Contradiction Method

• Proof by contradiction assumes the opposite of what we want to prove
• If this assumption leads to a logical contradiction, the original statement must be true
• Widely used in mathematics for proving the irrationality of numbers
• Steps involve assuming the number is rational, deriving a contradiction, and concluding irrationality
• Powerful technique for proving statements about infinite sets or abstract concepts

Demonstrating the Irrationality of √2

• Assume √2 is rational, can be expressed as $\frac{p}{q}$ where p and q are integers with no common factors
• Square both sides: $2 = \frac{p^2}{q^2}$
• Multiply by q²: $2q^2 = p^2$
• p² must be even, so p must be even (p = 2k for some integer k)
• Substitute: $2q^2 = (2k)^2 = 4k^2$
• Divide by 2: $q^2 = 2k^2$
• q² must be even, so q must be even
• Contradicts the assumption that p and q have no common factors
• Therefore, √2 must be irrational

Exploring Incommensurable Lengths Geometrically

• Incommensurable lengths lack a common unit of measurement
• Diagonal of a unit square has length √2, incommensurable with the side length
• Attempt to find a common measure leads to an infinite process of smaller and smaller squares
• Relates to the Euclidean algorithm for finding greatest common divisors
• Provides a geometric intuition for the irrationality of √2
• Extends to other irrational lengths in geometry (golden ratio in regular pentagons)