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,
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...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.
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.
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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
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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.
3) The Summa Theologica of St. Thomas Aquinas. Whether God Exists? 1920. New Advent. http://www.newadvent.org/summa/100203.htm. K. Knight. 2003.
Generally the laws of thought are considered the basis for any thought, discourse or discussion. They cannot be proved or disproved and to deny them is self-contradictory. It is widely known that there are mathematicians who d...
Taylor, C. Rationality in Martin Hollis and Steven Lukes editors Rationality and Relativism (Cambridge Press, 1982)
1.01100 = 2.7048 (1+1/1000)1000 = 1.0011000 = 2.7169 (1+1/10000)10000 = 1.000110000 = 2.7181 (1+1/) = (1+) = 2.7182818284590452353602874713526624977572470936. = e. Works Cited http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/. http://oakroadsystems.com/math/loglaws.htm http://www.physics.uoguelph.ca/tutorials/LOG/ http://en.wikipedia.org/wiki/Michael_Stifel http://en.wikipedia.org/wiki/John_Napier http://www.ndt-ed.org/EducationResources/Math/Math-e.htm http://www.thocp.net/reference/sciences/mathematics/logarithm_hist.htm http://mathforum.org/dr.math/faq/faq.pi.html
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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.
_ _ _. Letters to Bertrand Russell. Ed. Harry T. Moore. New York: Gotham Book Mart, 1948.
Ramasinghe, W. (2005). A Simple Proof e2 is Irrational. International Journal of Mathematical Education in Science and Technology, 36(4):407-441