Irrationality of exp(x)

In one of the earlier posts we indicated that Johann H. Lambert proved the irrationality of $\exp(x)$ or $e^{x}$ for non-zero rational $x$ by means of continued fraction expansion of $\tanh x$. In this post we provide another proof for irrationality of $e^{x}$ which is based on a completely different approach. I first read this proof from Carl Ludwig Siegel's wonderful book Transcendental Numbers and I was amazed by the simplicity and novelty of Siegel's argument.

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Irrationality of ζ(2) and ζ(3): Part 2

In the last post we proved that $\zeta(2)$ is irrational. Now we shall prove in a similar manner that $\zeta(3)$ is irrational. Note that this proof is based on Beukers' paper "A Note on the Irrationality of $\zeta(2)$ and $\zeta(3)$."

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Irrationality of ζ(2) and ζ(3): Part 1

Introduction

In 1978, R. Apery gave a mathematical talk which stunned the audience (consisting of other fellow mathematicians). Apery presented a very short proof of the irrationality of $\zeta(3)$ which created utter confusion and many believed his proof to be wrong. However some months later a few other mathematicians (primarily Henri Cohen) verified Apery's proof and concluded that it was correct.

Shortly after all this drama regarding Apery's proof, F. Beukers published another proof of irrationality of $\zeta(3)$ which is much simpler and comprehensible compared to the proof given by Apery. In this series of posts we will provide an exposition of Beukers' Proof. The content of this series is based on Beukers' paper "A Note on the Irrationality of $\zeta(2)$ and $\zeta(3)$."

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Proof that e squared is Not a Quadratic Irrationality

This post is based on the paper "Addition a la note sur l'irrationnalité du nombre e" by Joseph Liouville which contains proof of the fact that $e^{2}$ is not a quadratic irrationality.

In previous posts I covered that 1) $e^{2}, e^{4}$ are irrational and 2) $e$ is not a quadratic irrationality. I now present the final chapter in this series namely the:

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Another Proof that e squared is Irrational

In the last post we used the multiplication by $n!$ trick to prove that $e$ is not a quadratic irrationality. In this post we will use same technique albeit in a direct fashion to show that $e^{2}$ is irrational.

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Proof that e is Not a Quadratic Irrationality

Introduction

There are numerous proofs commonly available online for the fact that the Euler's number $e$ is irrational. Then going further we find that $e$ is also a transcendental number which means that it can not be the root of a polynomial equation with integral coefficients and thereby transcends the powers of algebra in a sense. Again the proof that $e$ is transcendental is also available on various places online.

In this post I am going to present the proof that $e$ is not a quadratic irrationality. This is based on the paper "Sur l'irrationnalité du nombre e = 2.718..." by Joseph Liouville.

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Irrationality of π(PI): Lambert’s Proof Contd.

Irrationality of Continued Fractions

We have the following results about the irrationality of some continued fractions:
If $ a_{n}, b_{n}$ are positive integers then:
1) The infinite continued fraction $$\frac{b_{2}}{a_{2} +}\,\frac{b_{3}}{a_{3} +}\,\frac{b_{4}}{a_{4} +}\,\frac{b_{5}}{a_{5} +}\,\cdots $$ converges to an irrational value, provided that $ a_{n} \geq b_{n}$ for all values of $ n$ starting from a certain value $ n = n_{0}$.

2) The infinite continued fraction $$\frac{b_{2}}{a_{2} -}\,\frac{b_{3}}{a_{3} -}\,\frac{b_{4}}{a_{4} -}\,\frac{b_{5}}{a_{5} -}\,\cdots $$ converges to an irrational value, provided that $ a_{n} \geq b_{n} + 1$ for all values of $ n$ starting from a certain value $ n = n_{0}$ and the condition $ a_{n} > b_{n} + 1$ must hold for an infinite number of values of $ n$.

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Irrationality of π(PI): Lambert's Proof

Introduction

After mentioning about the Lambert's famous proof of irrationality of $ \pi$ in an earlier post, it is now time to give it to the readers in its entirety. I need to reiterate the fact that being a far more direct proof than the modern proofs of Ivan Niven, it is still highly neglected by modern authors and educators. The idea of the proof is really elementary but based on the concept of continued fractions which are now deleted from the high school mathematical syllabus. Why this topic is now left out is still unclear to me. One reason which I can guess of is that the manipulations of continued fractions are not so simple (compared to those of an infinite product or a series). The visible form of the continued fraction does not give any idea about its value unless we do the calculations.

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