05 March 2020

Numerical Infinities


    What is the Arithmetic of Infinity?​

    If we use base ten notation, we can specify numerical values to carry out calculations or to determine the size of certain collections. For instance, the specifications 3 and 4 allow us to carry out certain computations like 3 + 4 or 4 - 3. They also allow us to count the items in certain collections: thus, the collection {0, 1, 2, 3} is determined by the specification 4 and, deleting the item 0 from it, we obtain the collection {1, 2, 3}, determined by the specification 3. ​

    As long as we stick to base ten, we cannot use any numerical specification to determine an infinite collection like {1, 2, 3, 4, ...} as distinct from the infinite collection {2, 3, 4, ....}, obtained from the former by deleting the item 1. When we attempt to assign determinations to such collections, we often identify them. We take them to be infinite, instead of registering their differences, and assign them the same specification 'infinity'. In this case, however, we prevent ourselves from carrying out calculations with infinity, since we end up with expressions like 'infinity - infinity', which do not lead us to any numerical value. ​

    The Arithmetic of Infinity is a way of remedying this inconvenience, by allowing us to work with a different notation in an infinite base: the new base is a kind of unit of measure, intended to determine the collection of all positive integers {1, 2, 3, ...}. With this new base, called 'gross-one', we can determine infinite collections and use infinite numerical specifications to carry out calculations as usual. ​

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    This website is a venue for students and teachers to learn about the Arithmetic of Infinity, a form of numerical analysis introduced by the Russian mathematician Yaroslav Sergeyev in 2003…​


    Numerical Infinity and the Infinity Computer: this is the official page listing ongoing research based on the Arithmetic of Infinity, as well as the academic prizes and acknowledgements received by its founder, Yaroslave Sergeyev​


    Association of Mathematics Teachers Officla Website (UK)​

    The Scottish Mathematical Council​

    N-Rich (Cambridge)​