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Paramanand's Math Notes

Be the first to leave a comment! 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 Siegels wonderful book Transcendental Numbers. And I was amazed by the simplicity and novelty of Siegels argument. And thi...

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Be the first to leave a comment! 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 Siegels wonderful book Transcendental Numbers. And I was amazed by the simplicity and novelty of Siegels argument. And thi...

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Paramanand's Math Notes | paramanands.blogspot.com Reviews

https://paramanands.blogspot.com

Be the first to leave a comment! 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 Siegels wonderful book Transcendental Numbers. And I was amazed by the simplicity and novelty of Siegels argument. And thi...

INTERNAL PAGES

paramanands.blogspot.com

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June 2013 | Paramanand's Math Notes

http://www.paramanands.blogspot.com/2013_06_01_archive.html

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. Friday, June 28, 2013. Congruence Properties of Partitions: Part 2. While stu...

2

Values of Rogers-Ramanujan Continued Fraction: Part 2 | Paramanand's Math Notes

http://www.paramanands.blogspot.com/2013/09/values-of-rogers-ramanujan-continued-fraction-part-2.html

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. Next note that to go from the second line to the third line we need to see that the powers of $q$ in the first pro...

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Irrationality of π(PI): Lambert’s Proof Contd. | Paramanand's Math Notes

http://www.paramanands.blogspot.com/2011/05/irrationality-of-pi-lamberts-proof-contd.html

Irrationality of π(PI): Lambert’s Proof Contd. Be the first to leave a comment! 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:. The infinite continued fraction. The infinite continued fraction. A {1}, A {2}$ are integers. Then we have $ 0 A {2} A {1}$. Irrationality of $ pi$. It is now easy to deduce the irrationality of $ pi$. We start with the following result:. And set $ x = p / q$...

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Congruence Properties of Partitions: Part 1 | Paramanand's Math Notes

http://www.paramanands.blogspot.com/2013/06/congruence-properties-of-partitions-part-1.html

Congruence Properties of Partitions: Part 1. Be the first to leave a comment! 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. Is denoted...

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September 2013 | Paramanand's Math Notes

http://www.paramanands.blogspot.com/2013_09_01_archive.html

Values of Rogers-Ramanujan Continued Fraction: Part 3. Be the first to leave a comment! 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$. We have from an earlier post. Monday, September 16, 2013. Monday, September 9, 2013.

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Pandy: Angle Sum Formula for Polygons

http://paramanand.blogspot.com/2010/12/angle-sum-formula-for-polygons.html

Not so cool stuff about everything! Thursday, December 02, 2010. Angle Sum Formula for Polygons. We are all familiar with the fact that the sum of all the angles of a triangle is 180 degrees. And for quadrilaterals the sum of its angles comes to be 360 degrees. This can be generalized to prove that the sum of angles of polygon of n. Sides is (n - 2) * 180 degrees. Second approach is to take a point in the interior of the polygon and join this point with every vertex of the polygon. This gives us n. Thus ...

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New Blog on Blogger | Notes Have Moved

https://paramanands.wordpress.com/2013/01/28/new-blog-on-blogger

Skip to search - Accesskey = s. New Blog on Blogger. By paramanands on January 28, 2013. I have created a new blog on blogger. Instead of the WordPress. Engine. The advantage of MathJax is that it produces scalable images (either made by special fonts or in SVG format) corresponding to the. Commands rather than plain static images generated by WordPress. All the existing posts on WordPress have now been moved to the new blog and any new entries will appear only in the new blog. Leave a Reply Cancel reply.

paramanand.blogspot.com

Pandy: Kudos to NCERT

http://paramanand.blogspot.com/2008/05/kudos-to-ncert.html

Not so cool stuff about everything! Thursday, May 15, 2008. For those not of the Indian origin, NCERT. National Council of Educational Research and Training) is a government body which publishes textbooks for students upto 12th grade apart from doing 'educational research and training'. Well to my utter excitement these NCERT books have improved their standard much beyond what I had seen in my time (1995-1998). First let me remark on the physical design (well, because it is remarkable enough! But the rea...

paramanand.blogspot.com

Pandy: Are Smartphones Fast Enough?

http://paramanand.blogspot.com/2013/07/are-smartphones-fast-enough.html

Not so cool stuff about everything! Saturday, July 20, 2013. Are Smartphones Fast Enough? For the last few years I have been writing a series of posts on my maths blog. When I started my blog I hosted it on Wordpress which used a LaTeX engine to support math content. Then I moved back to blogger which allowed full control of the template. I now use MathJax. For all the math content on my blog. Smartphones are not Fast Enough. Please think hard before spending huge sums of money on the latest smartphone!

paramanand.blogspot.com

Pandy: August 2006

http://paramanand.blogspot.com/2006_08_01_archive.html

Not so cool stuff about everything! Tuesday, August 29, 2006. Infinity Causes Infinite Confusion. There were some posts by people were engineering graduates and I was surprised to find that they still had similar feelings about the beast. I don't think that is excusable because almost all engineering courses have some course in basic calculus. But anyway, the real culprits are the teachers, professors and worst of all, the book authors who fail to explain even the very basic concepts like infinity. When ...

paramanand.blogspot.com

Pandy: June 2006

http://paramanand.blogspot.com/2006_06_01_archive.html

Not so cool stuff about everything! Saturday, June 03, 2006. Microsoft vs. Adobe. Recently Adobe is making a lot of uproar about suing Microsoft for bundling PDF export and XPS document format in Office 2007 and Windows Vista. XPS (XML Paper Specification) is Microsoft's answer to Adobe PostScript format and as Microsoft is saying is going to be a better technology. Why doesn't Adobe beat Microsoft in terms of quality rather than suing them? They created an open standard for widespread adoption by users ...

paramanand.blogspot.com

Pandy: September 2005

http://paramanand.blogspot.com/2005_09_01_archive.html

Not so cool stuff about everything! Tuesday, September 27, 2005. To Hell with Intelligent Design. Today I came to read a news article. I wonder how can a developed country like USA be so irrational enough to allow such stuff in schools. This is sharp contrast to their engineering and technological achievements and it bothers me too much. I think that ideas propounded in Bible and other similar literature should be a part of some subject like "Moral Science" or history or pure literature, but in no way sh...

paramanand.blogspot.com

Pandy: November 2006

http://paramanand.blogspot.com/2006_11_01_archive.html

Not so cool stuff about everything! Saturday, November 18, 2006. Book Review: Breaking the Spell. The book "Breaking the Spell: Religion as a Natural Phenomenon" is a very bold attempt by the renowned philosopher Daniel C. Dennett. It tries to study the phenomenon of religion from an objective viewpoint using the scientific methods of observations and logical reasoning. Like all other books by Dennett. It is highly thought provoking and I simply could not put the book down for 3 days. A religion without ...

paramanand.blogspot.com

Pandy: Another Proof of Pythagoras Theorem

http://paramanand.blogspot.com/2010/11/another-proof-of-pythagoras-theorem.html

Not so cool stuff about everything! Monday, November 29, 2010. Another Proof of Pythagoras Theorem. Pythagoras Theorem is one of the most important theorems in elementary geometry. It states that in a right angled triangle the square of the largest side is the sum of the squares of the other two sides. There are a large number of proofs available in textbooks and online articles. I will show here a very simple proof that uses only two concepts:. 1) Area is additive. 2) Congruent figures have same area.

paramanand.blogspot.com

Pandy: December 2005

http://paramanand.blogspot.com/2005_12_01_archive.html

Not so cool stuff about everything! Thursday, December 22, 2005. Those who have been following my blog for the last few months would have come across my article " To Hell With Intelligent Design. They would be glad to know that finally the ID theory has been sent to hell. Thanks to judge John Jones of Pennsylvania, who has banned the teaching of intelligent design at schools in Pennsylvania. For more details read this news article. Posted by Paramanand Singh at 1:06 PM. Friday, December 02, 2005. Which c...

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Paramanand's Math Notes

Be the first to leave a comment! 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 Siegels wonderful book Transcendental Numbers. And I was amazed by the simplicity and novelty of Siegels argument. And thi...

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Skip to search - Accesskey = s. New Blog on Blogger. By paramanands on January 28, 2013. I have created a new blog on blogger. Instead of the WordPress. Engine. The advantage of MathJax is that it produces scalable images (either made by special fonts or in SVG format) corresponding to the. Commands rather than plain static images generated by WordPress. All the existing posts on WordPress have now been moved to the new blog and any new entries will appear only in the new blog. Older Posts ».

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