to have various combinations of quantities and to construct terms having a more complex structure. In this paper, we are interested only on numbers having integer grosspowers.

Sums having a fixed infinite number of items and\
integrals over fixed infinite domains {#s3_riem}
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Introduction of the new numeral, 1, allows us to distinguish within $\infty$ many different infinite numbers in the same way as the introduction of modern numeral systems used to express finite numbers allows us to distinguish numbers ‘three’ and ‘four’ within ‘many’.

Thus, the Aristotle principle ‘The part is less than the whole’ leading to the fact that $x+1>x$ can be applied to all numbers $x$ (finite, infinite, and infinitesimal) and not only to finite numbers as it is usually done in the traditional mathematics (this point is discussed in detail in [@informatica; @Lagrange]). In general, we have that $x + y> x,\,\, y>0,$ where both $y$ and $x$ can be finite, infinite, or infinitesimal. For instance, it follows $$\G1+1 > \G1, \hspace{1cm} 1+\G1^{-1} > 1,
\hspace{1cm}\G1^2+1 > \G1^2, \hspace{1cm} \G1^2+\G1^{-1} >
\G1^2.$$

This means that such records as $S=a_1+a_2+\ldots$ or $\sum_{i=1}^{\infty}a_i$ become unprecise in the new mathematical language using grossone. By continuation the analogy with Pirahã 