Irrationality Proofs

Number theory has always been a fascinating topic to me. If one stops and thinks, the concept of irrational numbers is a very intriguing one; numbers that we cannot determine the exact value of. For example we cannot locate the exact value of √2 on the number line. I find this idea very strange and curious.

To be more precise, irrational numbers are those that cannot be expressed as the ratio of two integer numbers. One interesting approach is to examine irrational numbers by continued fractions. Every number can be written as a continued fraction in the form of

This is an alternative to decimal representations, which is not based on a specific base number. For example:

Irrational numbers are in fact produced from infinite continued fractions. For example:

The value of a number can be approximated to different accuracies by truncating the continued fraction at different poinst.

Determining the continued fraction for an irrational number is also very interesting and I might cover it in a later post. In this post, I am going to prove the irrationality of some numbers such as √2 and π.

Irrationality of √2:

This is proof by contradiction. Let’s assume there is a simplified fraction a/b which is equal to √2.

a/b=√2

a=b√2

a^2=2b^2

∴a^2 | 2

∴a|2 since a∈Z

Since a is even, b cannot be even (since a/b is a simplified fraction).

Since a is even, there should be an integer k for which a=2k.

(2k)^2=2b^2

4k^2=2b^2

b^2=2k^2

∴b^2 |2

∴b|2 since b ∈Z

But b cannot be even. Since a and b are both even, a/b is not a simplified fraction. Therefore, by proof by contradiction, √2 is irrational. Specifically, this is called proof by infinite descent.

Irrationality of π:

This proof is much more difficult and interesting. We are again going to use proof by contradiction. Since this proof involves a lot of exponents, derivatives, and integrals, I have written the proof in Microsoft Word and I am going to upload it here as images.

In future posts I might show irrationality proofs of other numbers as well as proof of transcendence, which is another interesting property of some numbers.