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Arithmetic of Infinity by [Sergeyev, Yaroslav D.]
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Length: 112 pages

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This book presents a new type of arithmetic that allows one to execute arithmetical operations with infinite numbers in the same manner as we are used to do with finite ones. The problem of infinity is considered in a coherent way different from (but not contradicting to) the famous theory founded by Georg Cantor. Surprisingly, the introduced arithmetical operations result in being very simple and are obtained as immediate extensions of the usual addition, multiplication, and division of finite numbers to infinite ones. This simplicity is a consequence of a newly developed positional numeral system used to express infinite numbers. In order to broaden the audience, the book was written as a popular one. This is the second revised edition of the book (the first paperback edition has been published in 2003).

Yaroslav D. Sergeyev, Ph.D., D.Sc., D.H.C. is Distinguished Professor at the University of Calabria, Italy and Professor at N.I. Lobachevski Nizhniy Novgorod University, Russia. He has been awarded several national and international research awards (Pythagoras International Prize in Mathematics, Italy, 2010; Outstanding Achievement Award from the 2010 World Congress in Computer Science, Computer Engineering, and Applied Computing, USA; Lagrange Lecture, Turin University, Italy, 2010; MAIK Prize for the best scientific monograph published in Russian, Moscow, 2008, etc.). His list of publications contains more than 200 items, among them 5 books. He is a member of editorial boards of 4 international scientific journals. He has given more than 30 keynote and plenary lectures at prestigious international congresses in mathematics and computer science. Software developed under his supervision is used in more than 40 countries of the world. Numerous magazines, newspapers, TV and radio channels have dedicated a lot of space to his research.

Opinions of some experts with respect to the new approach to infinity and the book are given below.
“Mathematicians have never been comfortable handling infinities, such as those that crop up in the area of a Sierpinski carpet. But an entirely new type of mathematics looks set to by-pass the problem”, MIT Technology Review, 03.19.2012.
" We will mention here the timely proposal of an enlarged numerical system advanced recently by Yaroslav D. Sergeyev. This is simpler than non standard enlargements in its conception, it does not require infinitistic constructions and affords easier and stronger computation power.” Lolli G. Infinitesimals and infinites in the history of mathematics: A brief survey, Applied Mathematics and Computation, 2012, 218(16), 7979–7988.
“He shows that it is possible to effectively work with infinite and infinitesimal quantities and to solve many problems connected to them in the field of applied and theoretical mathematics.” De Cosmis S., De Leone R. The use of Grossone in Mathematical Programming and Operations Research, Applied Mathematics and Computation, 2012, 218(16), 8029-8038.
“I am sure that the new approach presented in this book will have a very deep impact both on Mathematics and Computer Science.” From the review written by D. Trigiante in Computational Management Science, 2007, 4(1), 85-86.
“These ideas and future hardware prototypes may be productive in all fields of science where infinite and infinitesimal numbers (derivatives, integrals, series, fractals) are used.” From the review written by A. Adamatzky, Editor-in-Chief of the International Journal of Unconventional Computing, 2006, 2(2), 193-194.
“The expressed viewpoint on infinity gives possibilities to solve new applied problems using arithmetical operations with infinite and infinitesimal numbers that can be executed in a simple and clear way.” From the review written by P.M. Pardalos, Editor-in-Chief of the Journal of Global Optimization, 2006, 34, 157–158.

At the web page of the author, the interested reader can find a number of reviews and technical articles of several researches.

Product Details

  • Format: Kindle Edition
  • File Size: 1912 KB
  • Print Length: 112 pages
  • Publisher: Yaroslav D. Sergeyev; 2 edition (Nov. 12 2013)
  • Sold by: Amazon Digital Services LLC
  • Language: English
  • ASIN: B00G7RB1FS
  • Text-to-Speech: Enabled
  • X-Ray:
  • Word Wise: Not Enabled
  • Enhanced Typesetting: Not Enabled
  • Average Customer Review: Be the first to review this item
  • Amazon Bestsellers Rank: #1,057,575 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
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Most Helpful Customer Reviews on (beta) HASH(0x9a89c87c) out of 5 stars 3 reviews
5 of 5 people found the following review helpful
HASH(0x9a6f3cb4) out of 5 stars Interesting but not complete Dec 18 2013
By David Williams - Published on
Format: Kindle Edition Verified Purchase
Description: Author poses a method of expression of regular inverted infinitesimals (infinities) using the regular base system and infinite place-holders. The writing is basically elementary and conversational, not mathematically rigorous. This is not non-standard analysis or a normal analysis of non-invertible infinitesimals. It is a method of infinity-record-keeping which may be useful.

1. Very elementary and conversational.
2. Interesting and enjoyable read. Some good historical notes.
3. Handy method of keeping track of infinite values (or infinitesimals)

1. Very elementary.
2. No mathematical rigor.
3. Does not actually attempt to model or discuss the real problems posed by infinite and infinitesimal values.
4 of 5 people found the following review helpful
HASH(0x9a6a6258) out of 5 stars A book that dares to explain one of the greatest scientific breakthrough of the last millennium Dec 8 2013
By Sergey Loy - Published on
Format: Kindle Edition Verified Purchase
`Arithmetic of Infinity' is a book that dares to explain in popular way one of the greatest scientific breakthrough of the last millennium -- enhancement of the modern mathematics with a method of arithmetical operation over infinite and infinitesimal numbers. The author of this method -- Professor Yaroslav D. Sergeyev -- has coined it as 'Arithmetic of Infinity' awarded with Pythagoras International Prize for Mathematics in 2010.

The book comprises of three chapters starting with introduction to a commonly accepted viewpoint on real numbers rooted from a principle of Ancient Greeks that `the part is less than the whole'.

The second chapter reflexes on flaws of the commonly accepted point of view leading to numerous paradoxes and confusion to distinct numbers from their names (numerals) etc. As far as our understanding of the Universe relays on its mathematical model, we do give up getting in areas described with infinite and infinitesimal numbers. As centuries ago, when popular model of the Universe was depicted with the Earth plane held on backs of the gigantic turtles, most of the public still feel safer by sticking with imperfect but convenient model where deep space and human's body inner space continue to be treated as 'no-go zone'. For instance, it is obvious that a human body comprises of finite number of cells. However, current mathematics and computing fail on counting those cells not to mention on modelling their interaction.

'Arithmetic of Infinity' has made new horizons of the Universe being feasible. The last chapter of the book demonstrates this by giving elegant but detailed answers to questions and paradoxes baffled mathematicians for centuries.

Well, the future has arrived. Welcome in!
2 of 3 people found the following review helpful
HASH(0x9b3641ac) out of 5 stars Brilliant New Concept of Handling Infinity Numerically March 14 2014
By numericalsciences - Published on
Format: Kindle Edition
Professor Sergeyev has introduced us to a brilliant new concept to handle computations with infinity and infinitesimals. The book starts out with a very lucid explanation of the Cantor theory on countability and uncountability. The diagrams and examples are excellent and make this subject accessible to most. The book then goes on to its main focus, to introduce the Infinite Unit Axiom, Grossone and an infinite base number system where computations with infinite (and infinitesimal) numbers are possible. Added to all this, the text even contains some nice historical information on number systems and early mathematics. A truly recommended text and now in kindle format. Bravo!