Abel and the Insolvability of the Quintic: Part 1

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Most of the students come across the solution of linear and quadratic equations in their secondary classes. While the solution of a linear equation $ax + b = 0$ with $a, b$ being rational does not present any difficulties (because the solution $x$ itself turns out to be a rational number), a quadratic equation of the form $ax^{2} + bx + c = 0$ (with $a, b, c$ rational) does present significant challenges. For one thing the solution may not be rational and sometimes may not be even real. Usually one encounters the use of square roots to solve such an equation. Fortunately there is a standard formula for solving such equations $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$ so that the equation can be solved directly in terms of its literal coefficients.

Many mathematicians tried to extend these ideas to solve the equations of third and fourth degrees. Thus during the 16th century Cardano solved the cubic and Ferrari solved the quartic equation. Later in the 18th century Lagrange published his classic work "Reflexions sur la resolution algébrique des equations" in which he unified the existing methods of solving equations upto degree $4$. He hoped that unifying all the available approaches into one coherent theory would help in solving higher degree equations. But neither Lagrange nor any other mathematician was able to provide a solution to quintics (equations of degree $5$) or higher degree equations. Then in 1824 a young Nowergian mathematician Niels Henrik Abel proved that it is not possible to solve a quintic equation in the same way as it is possible to solve equations of degree $2, 3$ or $4$.

Teach Yourself Limits in 8 Hours: Part 4

After dealing with various techniques to evaluate limits we now provide proofs of the results on which these techniques are founded. This material is not difficult but definitely somewhat abstract and may not be suitable for beginners who are more interested in learning techniques and solving limit problems. But those who are interested in the justification of these techniques must pay great attention to what follows.

Proofs of Rules of Limits

We provide proofs for some of the rules and let the reader provide proofs for remaining rules based on similar line of argument. First we start with rule dealing with inequalities:
If $f(x) \leq g(x)$ in the neighborhood of $a$ then $\lim_{x \to a}f(x) \leq \lim_{x \to a}g(x)$ provided both these limits exist.

Teach Yourself Limits in 8 Hours: Part 3

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In last two posts we have developed basic concepts and rules of limits. Continuing our journey further we now introduce certain powerful tools which help us in evaluation of limits of complicated expressions. We start with the simplest technique first.

Limits using Logarithms

In case we need to evaluate the limit of an expression of type $\{f(x)\}^{g(x)}$ then we can take logarithm and then the evaluation of limits becomes simpler. We will first illustrate the technique through an example and then provide the justification.

Teach Yourself Limits in 8 Hours: Part 2

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After the definitions and basic examples in Part 1, we now focus on the rules of evaluation of limits which will be highly useful in solving various limit problems. We will postpone the proofs of these rules to the last post in the series to avoid any distraction.

Rules of Limits

In the following rules we assume that the functions described are defined in a certain neighborhood of $a$ except possibly at point $a$. All the relations between the functions (if any) also hold in this neighborhood of point $a$ (except possibly at point $a$).

Teach Yourself Limits in 8 Hours: Part 1



While looking at certain limit problems posed in math.stackexchange.com (henceforth to be called MSE) I found that most beginners studying limits are living in a fantasy world consisting of vague notions, infinities and what not. I too had my share of such experiences during my time as a student learning calculus but I was lucky enough to get over with this phase very quickly through the help of "A Course of Pure Mathematics".

Regarding the answers posted on MSE I found that most of the answers although correct were not suitable for beginners studying limits. Some answers suggested that their authors themselves had the same vague notions but they somehow managed to avoid their pitfalls. Some other answers were using sophisticated techniques which involved deeper concepts than the concept of limits itself. And there were some heated arguments favoring one approach over another.

Therefore I decided to write a series of posts providing a step by step approach to solving limit problems encountered in an introductory calculus course. I have tried to split the whole topic into $4$ posts and I believe that the gist of each post can be assimilated in not more than $2$ hours and that's the logic behind the title of this series.

Cavalieri's Principle and its Applications



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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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 $\zeta(3)$."

Values of Rogers-Ramanujan Continued Fraction: Part 3

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Evaluation of $R(e^{-2\pi/5})$

In the last post we established the transformation formula $$\left[\left[\frac{\sqrt{5} + 1}{2}\right]^{5} + R^{5}(e^{-2\alpha})\right]\left[\left[\frac{\sqrt{5} + 1}{2}\right]^{5} + R^{5}(e^{-2\beta})\right] = 5\sqrt{5}\left[\frac{\sqrt{5} + 1}{2}\right]^{5}\tag{1}$$ under the condition $\alpha\beta = \pi^{2}/5$.

If we put $\alpha = \pi$ then $\beta = \pi/5$ and since we already know the value of $R(e^{-2\pi})$ we can use equation $(1)$ to evaluate $R(e^{-2\pi/5})$. But in order to do that we need to calculate $R^{5}(e^{-2\pi})$ first.

We have from an earlier post $$R(e^{-2\pi}) = \sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{\sqrt{5} + 1}{2}$$

Values of Rogers-Ramanujan Continued Fraction: Part 2

Continuing our journey from the last post we will deduce further properties of the Rogers-Ramanujan Continued Fraction $R(q)$ which will help us to find out further values of $R(q)$. In this connection we first establish an identity concerning powers of $R(q)$.

Identity Concerning $R^{5}(q)$

Using the identity $(3)$ from the last post, Ramanujan established another fundamental property of $R(q)$ namely: $$\frac{1}{R^{5}(q)} - 11 - R^{5}(q) = \frac{f^{6}(-q)}{qf^{6}(-q^{5})}\tag{1}$$

Values of Rogers-Ramanujan Continued Fraction: Part 1

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A Wild Theorem by Ramanujan

In his letter dated 16th January 1913 to G. H. Hardy, Ramanujan presented the following wild theorem: $$\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1 + \cfrac{e^{-6\pi}}{1 + \cdots}}}} = \left(\sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{\sqrt{5} + 1}{2}\right)\sqrt[5]{e^{2\pi}}\tag{1}$$ The theorem looks so strange and surprising, coming out of nowhere that Hardy had to remark: "they must be true because, if they were not true, no one would have had the imagination to invent them." In this post we will prove the above theorem using elementary methods. The proof is essentially the one given by Watson who claimed that probably Ramanujan obtained the result in the same manner.

Fundamental Theorem of Algebra: Two Proofs



Fundamental theorem of algebra is one of the most famous results provided in higher secondary courses of mathematics. Normally it is mentioned in chapter related to complex numbers where the reader is made aware of the power of complex numbers in solving polynomial equations. The theorem guarantees that any non-constant polynomial with real or complex coefficients has a complex root.

Congruence Properties of Partitions: Part 2

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Continuing our journey of partition congruences from the last post we now prove the congruences modulo $7$ and $11$.

Congruence Properties of Partitions: Part 1

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We know that any positive integer greater than $1$ can be expressed as a product of prime numbers in a unique fashion ignoring the order of factors. This is one of the most basic results in number theory and is aptly called the fundamental theorem of arithmetic. This result also shows that prime numbers are the building blocks for all integers and this justifies the importance given to prime numbers in number theory.

Thoughts On Ramanujan

Of late I had been reading Ramanujan's Collected Papers and based on my understanding of it (and inputs from works of Borwein brothers, Bruce C. Berndt) I wrote a series of posts explaining some of Ramanujan's discoveries (see 10 posts starting from here and 4 posts beginning from here). While studying Ramanujan's Papers I could not help myself being astounded by the depth of his discoveries and the ingenuity of the proofs he provided for some of his results.

Reading Papers has not been an easy job for me and seems like an unending task if I wish to have a complete and thorough understanding of it. Hence I decided to take a break for sometime and dedicate one of my posts about my thoughts on Ramanujan, his works, abilities and methods. Needless to say whatever I present here would be a personal view and may differ from general perception a reader might have of Ramanujan and his works. Because of the same reason this post is bound to be of somewhat personal nature.

Proof of Chudnovsky Series for 1/π(PI)

In 1988 D. V. Chudnovsky and G. V. Chudnovsky (now famous as "Chudnovsky Brothers") established a general series for $\pi$ by extending Ramanujan's ideas (presented in this series of posts). It can be however shown that their general series can be derived using Ramanujan's technique. Chudnovsky's approach has the advantage that using class field theory the algebraic nature of parameters in the general series can be determined and this greatly aids in the empirical evaluation of the parameters and thereby providing an actual series consisting of numbers.

Certain Lambert Series Identities and their Proof via Trigonometry: Part 2

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In the last post we saw that using a trigonometric identity Ramanujan was able to express the functions $S_{2n - 1}(x)$ or equivalently $\Phi_{0, 2n - 1}(x)$ in terms of simpler functions $P(x), Q(x), R(x)$. Continuing our journey further we start with the equation: \begin{align}&\left(\frac{1}{4}\cot\frac{\theta}{2} + \frac{x\sin\theta}{1 - x} + \frac{x^{2}\sin 2\theta}{1 - x^{2}} + \frac{x^{3}\sin 3\theta}{1 - x^{3}} + \cdots\right)^{2}\notag\\ &\,\,\,\,= \left(\frac{1}{4}\cot\frac{\theta}{2}\right)^{2} + \frac{x\cos\theta}{(1 - x)^{2}} + \frac{x^{2}\cos 2\theta}{(1 - x^{2})^{2}} + \frac{x^{3}\cos 3\theta}{(1 - x^{3})^{2}} + \cdots\notag\\ &\,\,\,\,+\frac{1}{2}\left\{\frac{x(1 - \cos\theta)}{1 - x} + \frac{2x^{2}(1 - \cos 2\theta)}{1 - x^{2}} + \frac{3x^{3}(1 - \cos 3\theta)}{1 - x^{3}} + \cdots\right\}\tag{1}\end{align} which was established in the last post.

Certain Lambert Series Identities and their Proof via Trigonometry: Part 1

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This is yet another post based on a paper of Ramanujan titled "On certain arithmetical functions" which appeared in Transactions of the Cambridge Philosophical Society in 1916. In this paper Ramanujan provided a lot of identities concerning Lambert series and thereby deduced many relations between various divisor functions. Apart from the amazing results proved in this paper, what I liked most is the very elementary approach followed by Ramanujan compared to the methods of modern authors who are seduced by the modular form.

Rogers-Ramanujan Identities: A Proof by Ramanujan



The history of Rogers-Ramanujan identities is well described in various books and papers. In brief these identities were first discovered and proved by L. J. Rogers in 1894 and then later re-discovered (but not proved) by Ramanujan in 1913. Later in 1919 Ramanujan published a proof. It is this proof which will be described here.

Ramanujan was not used to publishing proofs of many of his discoveries and hence there is a feeling (even now) that his methods were mystical and often inspired by his dreams. However he did publish some proofs and when we study these it becomes at once very clear that Ramanujan possessed proofs of almost all the results he found, but it was just lack of time and resources due to which he did not record the proofs. Unfortunately this is a big loss for mathematics because from the nature of his formulas it seems that his methods were highly efficient and startling at the same time. The proof we present in this post also has the same qualities and I hope the reader will enjoy going through these.

Proof that e squared is Not a Quadratic Irrationality

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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:

Another Proof that e squared is Irrational

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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.

Proof that e is Not a Quadratic Irrationality

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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.