Mathematics · Free PDF
Every irrational number demands its own proof.
The divisibility argument that settles sqrt(2) does nothing against e, and neither touches pi. This free book collects five distinct methods for proving irrationality, each demonstrated with multiple examples.
Free PDF · instant download · no payment at checkout
Format:
What’s inside
Five methods, grouped by the numbers they crack.
Different numbers need different tools. The book is organized around the tools.
Divisibility arguments
- The classic sqrt(2) proof, presented first
- Generalization to sqrt(n) for any non-perfect square
- Irrationality of sqrt(2) + sqrt(3)
- Irrationality of log_2(3)
Power series methods
- Fourier’s proof that e is irrational, presented in full
- The irrationality of e^2
- The approach that handles transcendental numbers
Continued fractions
- Proofs and approximations from the same machinery
- Niven’s 1947 proof of the irrationality of pi, in its own chapter
Numbers proven irrational
- sqrt(2), by three different methods
- sqrt(n) for any non-perfect square
- e and e^2
- pi, via Niven’s proof
- The golden ratio phi
Each method is demonstrated with multiple examples, not just its headline number.
Look inside the book.
Preview the writing, structure, and page layout before you decide.
Get the book
Proofs of Irrationality
Free
Digital PDF · instant download after a free checkout
- Prove sqrt(2) irrational by three different methods
- Generalize the argument to sqrt(n) for any non-perfect square
- Work through Fourier’s proof that e is irrational, then e^2
- Follow Niven’s 1947 proof of the irrationality of pi
- Use continued fractions for both proofs and approximations
- Settle sqrt(2) + sqrt(3), log_2(3), and the golden ratio phi
No payment. Checkout creates your download link instantly.
Questions readers ask
Asked before downloading, answered straight.
Is it actually free?
Yes. The checkout has no payment step. It creates your order, issues the download link instantly, and that’s the whole transaction.
Why is there a checkout at all?
Because the download is delivered through my store instead of a raw file link. The free checkout creates your order and your download link instantly, with no payment details anywhere in the flow.
How much math do I need?
If you can follow a divisibility argument, the Pythagorean chapters are open to you. The power series and continued fraction methods ask a little more, and Niven’s 1947 proof of pi gets a chapter of its own.
Are the proofs written out, or just sketched?
Written out. Fourier’s proof of the irrationality of e is presented in full, and each of the five methods is demonstrated with multiple examples.
Where should I start?
The book opens with the classic proof for sqrt(2), then generalizes it to sqrt(n) for non-perfect squares. After that, follow whichever number you care about: e, pi, or the golden ratio.
What I’m not going to claim
There are no reader ratings to show you. This is a free PDF, and free PDFs don’t collect reviews, so I’d rather say that plainly than dress the page up.
It’s strictly about irrationality. Proving that e or pi is transcendental is a different and much harder job, and this book doesn’t attempt it.
And it’s organized around five methods and the numbers listed above. If a number isn’t on that list, the book doesn’t prove it.
Hey, I’m Gaurav 👋
I taught mathematics before I built websites, and this is the topic where students hit a wall: the sqrt(2) argument they’d memorized wouldn’t transfer to e or pi. This book is the toolbox I wanted to hand them, five methods instead of one trick.
The numbers, since you probably won’t read a full bio:
- 18 years building on the web, and a WordPress core contributor
- 2,000+ articles published since 2008, the earliest of them mathematics notes
- 850+ clients
Free download
Prove one tonight.
Pick the number that made you curious and read its chapter. The whole book costs you a click.
Free PDF · instant download · no payment at checkout