1 = 1 0 = 0, 1-1= 0, =1, 1\^0=1, 1\^=1, 0\^=0. \[3.2.1\]

Various numeral systems including 1 can be used for expressing infinite and infinitesimal numbers. In particular, records similar to traditional positional numeral systems can be used (see [@informatica; @Korea]). In order to construct a number $C$ in this system, we subdivide $C$ into groups corresponding to powers of grossone: C = c\_[p\_[m]{}]{} 1\^[p\_[m]{}]{} + …+ c\_[p\_[1]{}]{} 1\^[p\_[1]{}]{} +c\_[p\_[0]{}]{} 1\^[p\_[0]{}]{} + c\_[p\_[-1]{}]{} 1\^[p\_[-1]{}]{} + …+ c\_[p\_[-k]{}]{} 1\^[p\_[-k]{}]{}. \[3.12\] Then, the record C = c\_[p\_[m]{}]{} 1\^[p\_[m]{}]{} …c\_[p\_[1]{}]{} 1\^[p\_[1]{}]{} c\_[p\_[0]{}]{} 1\^[p\_[0]{}]{} c\_[p\_[-1]{}]{} 1\^[p\_[-1]{}]{} …c\_[p\_[-k]{}]{} 1\^[p\_[-k]{}]{} \[3.13\] represents the number $C$, where finite numbers $c_i\neq0$ called *grossdigits* can be positive or negative. They show how many corresponding units should be added or subtracted in order to form the number $C$. Grossdigits can be expressed by several symbols.

Numbers $p_i$ in (\[3.13\]) called *grosspowers* can be finite, infinite, and infinitesimal, they are sorted in the decreasing order with $ p_0=0$ $$p_{m} >  p_{m-1}  > \ldots    > p_{1} > p_0 > p_{-1}  > \ldots
p_{-(k-1)}  >   p_{-k}.$$ In the record (\[3.13\]), we write $\G1^{p_{i}}$ explicitly because in the new numeral positional system the number $i$ in general is not equal to the grosspower 