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Ganguli, Nirendra Mohan
- Classification of Irrational Numbers
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1 234A N. S. C. Bose Road, Kolkata 700040, IN
1 234A N. S. C. Bose Road, Kolkata 700040, IN
Source
Indian Science Cruiser, Vol 33, No 1 (2019), Pagination: 57-62Abstract
An irrational number expressed in decimal form / infinite continued fractions or sum of infinite series of rational numbers, can only evaluate its approximate magnitude, but cannot represent it. There are different kinds of irrational numbers. One of the kinds is generated by extraction of ischolar_mains of the positive rational numbers. The deliberation shows that all irrational numbers of this kind are not of similar category, which has been defined as the ‘order of irrational numbers’.
References
- R. Courant & H. Robbins, (edited by Stewart, I.): What is Mathematics? An Elementary
- Approach to Ideas and Methods, 1996, 2nd Edition, New York/Oxford: Oxford University Press.
- G. F. Simmons, ‘Introduction to Topology and Modern Analysis’, Tata McGraw Hills Edition (2004).
- David M. Burton, ‘Elementary Number Theory’, 1995, 2nd Edition, Universal Book Stall, New Delhi.
- An Alternate Simple Proof of Fermat’s Last Theorem
Abstract Views :331 |
PDF Views:3
Authors
Affiliations
1 Suryatoran, Flat No. 2B, 234A N. S. C. Bose Road, Kolkata 700040, IN
1 Suryatoran, Flat No. 2B, 234A N. S. C. Bose Road, Kolkata 700040, IN
Source
Indian Science Cruiser, Vol 33, No 6 (2019), Pagination: 22-25Abstract
ℜ denotes the real line as well as the set of all real numbers, and the real numbers are considered as if they are points on the real line, and the points on the real line as if they are real numbers. Hence, three positive rational numbers a, b & c, satisfying the condition cn = an + bn, where abc ≠ 0, and n is an integer greater than 1, will represent three sides of a triangle, say ΔABC, on a two dimensional plane. Consequently, the algebraic equation will yield the trigonometric equation SinnA + SinnB = Sinn(A+B). The solution of this equation, in general, will be in the form of B = f(A), where the relation between the variables A & B has to be linear in nature on account of the properties of the ΔABC, which is on a two dimensional plane with the relation between the vertical angles given by (A+B+C) = π, as well as the exponent of the variables A & B of the trigonometric equation is also one; but this condition is satisfied only when n = 2. Hence, the algebraic equation has rational solutions only with n = 2 and the FLT is true.Keywords
Rational Numbers, Real Line, Triangle, Trigonometric Equation.References
- Singh, Simon; Fermat’s Last Theorem, 1998(Paper Pack), Fourth Estate Limited, London.
- Burton, David M.: Elementary Number Theory, 1995(Reprint), Universal Book Stall, New Delhi.
- Courant, R., Robbins, H., and Stewart, I.; What is Mathematics? An Elementary Approach to Ideas and Methods, 1996, 2nd Edition, New York/Oxford: Oxford University Press.
- G. F., Simmons; Introduction to Topology and Modern Analysis, Tata McGraw-Hill Edition 2004, Tata McGraw-Hill Education Pvt. Ltd., Delhi.
- An Alternative Definition of Irrational Numbers
Abstract Views :276 |
PDF Views:0
Authors
Affiliations
1 Flat No. 2B, 234A N. S. C. Bose Road, Kolkata - 700040, IN
1 Flat No. 2B, 234A N. S. C. Bose Road, Kolkata - 700040, IN
Source
Indian Science Cruiser, Vol 36, No 1 (2022), Pagination: 48-50Abstract
An algebraic real number is irrational when the index of at least one of its prime factors is incongruent with respect to the desired ischolar_main of the number. In this article the author has discussed about an alternative definition of irrational numbers.Keywords
Numbers, Irrational Numbers, Definition, Alternative Definition.References
- I Nivan, Numbers: Rational and Irrational, Random House and The L. W. Singer Co., New York, 1961.
- R Courant and H Robbins, What is Mathematics: An Elementary Approach to Ideas and Methods, 2nd Edition, Oxford University Press, Oxford, 1996.
- Counting of Numbers
Abstract Views :115 |
PDF Views:0
Authors
Affiliations
1 Suryatoran, Flat No. 2B, 234A, N. S. C. Bose Road, Kolkata 700040, IN
1 Suryatoran, Flat No. 2B, 234A, N. S. C. Bose Road, Kolkata 700040, IN
Source
Indian Science Cruiser, Vol 37, No 1 (2023), Pagination: 33-38Abstract
‘χ0’ and ‘c’, termed as transfinite numbers, signify the number of elements in the infinite set of positive integers and that of real numbers respectively. Cantor proved that c > χ0 using the diagonal argument; but the infinitude of ‘c’ is on two counts i.e. magnitude and type of the element while that of χ0 is on single count i.e. magnitude. ‘c’ is not numerically greater than ‘χ0’, but denotes a higher degree of infinity. Hence the continuum hypothesis is rather irrelevant.Keywords
Natural numbers, Transfinite numbers, Order of irrational numbers, Degree of infinityReferences
- S Hawking, God Created the Integers, Penguin Books, UK, 2006.
- W C Salmon, Zeno’s Paradoxes, 2nd Edition, Indianapolis: Hackett Publishing Co. Inc., 2001.
- G F Simmons, Introduction to Topology and Modern Analysis, Tata McGrew-Hill, 2004.
- R Courant, H Robbins and I Stewart, What is Mathematics? An Elementary Approach to Ideas and Methods, 1996, 2nd Edition, New York/Oxford: Oxford University Press.
- N M Ganguli, Classification of Irrational Numbers, Indian Science Cruiser, Vol 33, No 1, page 57-62, 2019, Institute of Science, Education and Culture, Kolkata.