Cavalieri's Principle and its Applications

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Introduction

In this post we will discuss something is which is very elementary and fascinating, yet not available in a high school curriculum. More precisely we will study a part of solid geometry related to calculation of volume of solids. In so doing we will need the famous Cavalieri's Principle which relates volumes of two solids under certain conditions.

Cavalieri's Principle

The Cavalieri's Principle states that:
If two solids lie between two parallel planes and any plane parallel to these planes intersects both the solids into cross sections of equal areas then the two solids have the same volume.

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

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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 $\zeta(3)$."

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

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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 $\zeta(3)$."