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 (Khwarizmi International Award; Pythagoras International Prize in Mathematics; Lagrange Lecture, Turin University, Italy; MAIK Prize for the best scientific monograph published in Russian, Moscow, etc.). His list of publications contains more than 250 items, among them 6 books. He is a member of editorial boards of 6 international scientific journals. He has given more than 60 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.
