Review of " On the irrationality of π4 and π6 " by Md. Reza Yegan
INTRODUCTION
"On the irrationality of π4 and π6 " by Md. Reza Yegan, taken from the Journal of Number Theory is a paper that, quite simply put, explores the concept of irrationality of 2 specific powers of π, namely π4 and π6. Referencing other papers as examples, Yegan states that, though the irrationality of π and π2 are often discussed, the irrational nature of the higher powers of π are usually neglected. Hence, Yegan chooses to explore the irrational natures of 2 higher powers of π : π4 and π6 .The paper under review thus contains 2 simple proofs to explain the irrationality of π4 and π6 using integral identities (Hermite's technique) on a specific sequence of functions.
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Though he is successful in explaining, by contradiction, the irrationality of π4 and π6, he fails to find proof for a general case to test irrationality of all powers of pi, despite discussing the need to look into irrationality higher powers of pi due to their neglected existence. His overall explanation is convoluted and lengthy in comparison to other papers that have succeeded in giving shorter proofs, and have even found general cases. Thus, while Yeman's paper succeeds in doing what has set out to do, it fails to expand on the base it creates.
REFERENCES
[1] M.R. Yegan, On the irrationality of π4 and π6, J. Number Theory (2017), http://dx.doi.org/10.1016/j.jnt.2017.02.009 [2] Breusch, R., A proof of the irrationality of π, the American Mathematical Monthly , Vol.61,
NO.9 (Nov.1954), pp.631-632.
[3] Huylebrouck, D., Similarities in irrationality proofs for π, ln 2, ζ(2),and ζ(3),
AMM 108(2001) 222-231.
[4] Nesterenko, Yu., V., A simple proof of the irrationality of π, Russ. J. Math. Phys. 13(2006), no.4, 473.
[5] Niven, I., A simple proof that π is irrational, Bulletin of the American Mathematical
Anum Munaf Dr. Caryn Voskuil PHIL-1301-83456 23 April 2017 Response Paper: Chapter 2 In Chapter two “God and the Origin of the Universe” of the book “Problems from Philosophy”, written by James Rachels and Stuart Rachels, a very interesting and contentious topic has been discussed. It is about the belief that God exists in this universe or not and this universe is created by God or it has been developed by chance. Rachels with the help of distinct types of arguments tried to prove that God exists in this world and the universe is created by some intelligent designer. At the beginning, he gave the results of recent Gallop poll and Pew Research Center polls to explain that how many people are religious and how many are non-religious.
As Pi is an active disciple of three separate religions, one would assume he has a shifting opinion on reality and it’s roots. Despite seeing himself as a practicing Hindu, Christian, and Muslim, he believes that there is a unity of all things. This contradicts
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
In Martin Hollis and Steven Lukes editors Rationality and Relativism (Cambridge Press, 1982).
...ieve what I see” as the basis for all justification is unreasonable though because not everyone has seen every fact known to man. Simply believing everything Pi has told them would be irrational due to lack of scientific evidence. There is a lot more to prove that Pi’s condition just prompted him to create such an incredible story to deal with the immense tragedies he was put through.
» Part 1 Logarithms initially originated in an early form along with logarithm tables published by the Augustinian Monk Michael Stifel when he published ’Arithmetica integra’ in 1544. In the same publication, Stifel also became the first person to use the word ‘exponent’ and the first to indicate multiplication without the use of a symbol. In addition to mathematical findings, he also later anonymously published his prediction that at 8:00am on the 19th of October 1533, the world would end and it would be judgement day. However the Scottish astronomer, physicist, mathematician and astrologer John Napier is more famously known as the person who discovered them due to his work in 1614 called ‘Mirifici Logarithmorum Canonis Descriptio’.
Pi, short for Piscine, meaning a rational source of water, is a rational man living in the irrational world, who believes in not one, but three religions, which some may say is irrational. Pi, whose family owned a zoo, faced many hardships
By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.
3) The Summa Theologica of St. Thomas Aquinas. Whether God Exists? 1920. New Advent. http://www.newadvent.org/summa/100203.htm. K. Knight. 2003.
P = p1p2 ···pn+1. Let p be prime dividing P. If p is equal to pi for some i, then p divides
Even the smallest tasks can impact the world in a significant way. Math, despite its trivial appearance, is large in grandeur that governs our world from the inside and the outside. The many twists and turns that exist in Mathematics make its versatility unparalleled and continues to awe the many Mathematicians today and the many more to come. The Binomial Theorem is one such phenomenon, which was founded by the combined efforts of Blaise Pascal, Isaac Newton and many others. This theorem is mainly algebraic, which contains binomial functions, arithmetic sequences and sigma notation. I chose the Binomial Theorem because of its complexity, yet simplicity. Its efficiency fascinates me and I would like to share this theorem that can be utilized to solve things in the Mathematical world that seem too daunting to be calculated by normal means.
_ _ _. Letters to Bertrand Russell. Ed. Harry T. Moore. New York: Gotham Book Mart, 1948.
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the numbers, which is entirely his own independent discovery.
There are many people that contributed to the discovery of irrational numbers. Some of these people include Hippasus of Metapontum, Leonard Euler, Archimedes, and Phidias. Hippasus found the √2. Leonard Euler found the number e. Archimedes found Π. Phidias found the golden ratio. Hippasus found the first irrational number of √2. In the 5th century, he was trying to find the length of the sides of a pentagon. He successfully found the irrational number when he found the hypotenuse of an isosceles right triangle. He is thought to have found this magnificent finding at sea. However, his work is often discounted or not recognized because he was supposedly thrown overboard by fellow shipmates. His work contradicted the Pythagorean mathematics that was already in place. The fundamentals of the Pythagorean mathematics was that number and geometry were not able to be separated (Irrational Number, 2014).
To lead a pleasant and long-lasting life a person must find their source of a higher power, the higher power is used for guidance and to form morals along life’s course. As one is growing and their ethics are forming, reason finds its place alongside faith in one’s life. As reason comes to the surface a person must learn how to grasp and understand both concepts to be able to use them in important decisions. In Life of Pi the protagonist, Pi Patel, endures a series of tragic events, but it does not dawn on him that he must be cautious with every decision he makes. Instead of realizing the extremity of his situation, Pi uses his mind and creates a story to mask the madness of what is really happening. He uses this story to hide true feelings,