Proofs of Irrationality
Proving a number is irrational requires different tools depending on the number. This book covers five distinct methods, each demonstrated with multiple examples and varying levels of difficulty. Proofs of Irrationality provides a focused, proof-based study of irrationality that builds mathematical reasoning skills while exploring one of number theory's fundamental concepts.
Proving a number is irrational requires different tools depending on the number. This book covers five distinct methods, each demonstrated with multiple examples.
The Pythagorean approach uses divisibility arguments. The classic proof for sqrt(2) is presented first, followed by generalization to sqrt(n) for non-perfect squares, irrationality of sqrt(2) + sqrt(3), and irrationality of log_2(3).
Power series methods handle transcendental numbers. Fourier’s proof of the irrationality of e is presented in full, along with the irrationality of e^2.
Continued fractions provide both proofs and approximations. Niven’s 1947 proof of the irrationality of pi gets its own chapter.
Numbers Proven Irrational
- sqrt(2) via three different methods
- sqrt(n) for any non-perfect square
- e (Euler’s number) and e^2
- pi (Niven’s proof)
- The golden ratio phi